Some Strangeness in the Proportion
WE ARE JANUS-FACED. This is quite a strange fact about us. We owe our successful evolution in part to our religious proclivities, yet these same religious proclivities may kill us. So, for many of us, our two sides are at war (see chapter 9)—not only over the evidence for our religions, but over the moral problems religions create. One wonders if our evolution was worth the trouble. But this question is moot, now. We are here and we are going to be around at least long enough to utterly alter Earth and ourselves.
However, our being Janus-faced doesn’t mean that there isn’t something else out there—something that might be a worthy replacement for our religious strivings; something which our religious face might do well to turn toward. . . . There is indeed something out there. What it is, I cannot say. But I can point to it. That’s what this chapter is about.
I will discuss in detail three mysterious things: consciousness, infinity, and the rarity of the commonplace. My goal is to lay bare the strange nature of these three in such a way that you, the reader, can experience it in some depth. At the end of the chapter, I hope to have upset any blithe nonchalance you may have about these three seemingly ordinary things that surround you. And of course, these three are just the tip of the tip of the iceberg.
These mysteries are, I claim, the excellent beauties. And they reveal that our world is not flatly natural. I am not denying that the flatly natural worldview has room for beauty. It clearly does. There are beautiful explanations for such things as how flowers bloom or how dinosaurs lived their lives. And science reveals beautiful hidden processes like photosynthesis, pollination, and embryogenesis, for example. But the flatly natural view is committed to the notion that, at least ideally, science will leave nothing of importance unexplained. The flatly natural view insists that all scientific beauty must supervene on scientific understanding. But this is wrong. The excellent beauties show that there is far more to our world than what is revealed by scientific understanding. And yet it is science itself that reveals the excellent beauties. So, what we can see from our scientific vantage point greatly exceeds what we can comprehend from it.
The specific things I want to draw your attention to may strike you as irrelevant to a spiritual journey. But they are relevant, especially now after we’ve seen how utterly mundane and natural all our current religions really are. In chapters 11, 12 and 13, we will discuss why these phenomena are relevant to a journey that might be called “spiritual,” at least in some sense. For now, I encourage you to let these examples wash over you. And at the end of the chapter, try to see behind and beyond the specific details of the three examples to what they have in common, to what they point to. Let the mere existence of these examples be a source of wonder.
I want to stress that the mysteries I am going to discuss exist independently of both our religious face and our rational one. They exist objectively for all to see. The mysteries are real . . . and they are trying to tell us something.
The Conscious Self
What is consciousness? I don’t mean anything unusual or strange by the word “consciousness” (see chapter 1). Consciousness—or being conscious—is the most ordinary thing in your life . . . so ordinary, you rarely note it or think of it. Consciousness is the way the world seems to you, the way we experience it, feel it. Taste an apple, see a sunset, smell a rose or an angry skunk, stub your toe on the foot of the bed frame at 4:00 AM, hear your dog breathe or a baby gurgle and coo. These are conscious experiences. We have experiences because we are conscious. Or, rather, our having them constitutes our being conscious. Being conscious is what makes it fun or horrible or merely boring to be a human (or anything else that is conscious). Using a phrase that the philosopher Thomas Nagel made famous (among philosophers), we can say that a being is conscious when there is something it is like to be that being.1
The problems with consciousness begin right with its definition. To define consciousness, I can only point to what I assume are your conscious experiences. Or I can point to mine and assume that you have similar ones. That is, the only way I can define consciousness is by appealing to something I can only assume you have. There is no more-robust, scientific, third-person definition available. I have to say: “Look up at the cloudless sky. That color you are experiencing is blue. Your experience of blue is an example of being conscious, of conscious experience.” Or: “Stand on the beach. That repetitious, mild roaring sound you hear is the sound of waves coming in on the beach. Your experience of that roar is an example of being conscious, of conscious experience.” . . . That’s the best I, or anyone else, can do. This ostensive definition-plus-assuming is the same one researchers have been using to define consciousness since research of any sort was first conducted on it. And it is the same one consciousness researchers use today.
It’s all this assuming that is the problem. I can’t know what you in fact experience when you look up into the clear blue sky. Let’s contrast this with something else that is weird about consciousness, but that we do know exists via verbal reports. When you hear the waves hitting the beach, you may also see the sound as certain colors. For example, the sound of waves may also be experienced visually as, say, the color indigo. Some humans (and, for all we know, other animals) routinely have such mixed conscious sensory experiences. The phenomenon is called synesthesia.2 We know synesthesia exists because the people who experience it tell us. But here, as with nonsynesthetes like you (probably) and me, I have to assume that the synesthetes experience the sound of the waves as indigo. When they say, “I hear the waves and see this sound as indigo,” I have to take them at their word. I cannot know the actual content of the synesthete’s experience, just as I cannot know the content of your experiences, nor you mine.
This assuming leaves a lot of room for differences between us all. The color you see when you look at the sky may be the same color I see when I look at the green trees in a forest, or when I look at the rind of a grapefruit. This phenomenon is called color inversion and is quite different from synesthesia. For one thing, we have zero evidence that color inversion exists, and furthermore, it would be impossible to get any evidence. Suppose the colors I see are just the reverse of yours—our spectra are inverted, as the philosophers say. So I see the color yellow where you see blue. But we both look in the direction of the clear summer sky and say, “Blue.” We have both been taught that the color of the sky is picked out by the word “blue.” And we use that word regardless of what color we actually see. And I, like you, will say that the color of the sky is the same as the color of deep water: blue. But I will see yellow as the color of deep water. I will also say things like this: “Though they are called ‘blueberries,’ blueberries are not really blue like the sky but closer to purple like violets.” This is exactly what you say (give or take some arguing about the purpleness of blueberries—perhaps they are violet or indigo). But where you see blue and purple, I see yellow and a shade of red or orange. No one, and certainly not I, can know that my color spectrum is inverted relative to yours, perhaps relative to everyone’s. No report of any color experience I have can inform you or anyone else of our inverted spectra. In fact, this inversion could describe things way too simply. It could well be that every single human sees a different color when he or she looks into the clear noon sky.
At this point, you might ask, “Well, what about neuroscience or psychology? Can’t they help determine if we are all seeing blue when we look at the sky?” The answer is that neither is of any help at all. This is the essence of the strangeness and mystery of consciousness—psychology, neuroscience, whatever, cannot be of any help, ever. This claim is controversial, of course, but not as controversial as you might think. It is safe to say that most scientists working on consciousness are what philosophers call materialists or physicalists: most scientists think that consciousness, the self, conscious awareness, and a few other related phenomena are all the physical result of physical (or material) processes in our bodies; in particular, consciousness is thought to be due to neural processes in brains. Just as heat (like temperature in general) is the result of rapidly jostling molecules, consciousness is the result of the brain doing . . . ?? . . . something . . . (this list here of hypothesized somethings is disturbingly large, as we will see). Perhaps even you believe that consciousness is due to some physical brain process. Nevertheless, physicalism about consciousness is quite possibly false. And in any case, the evidence for it is far too weak to grant physicalism the status of a confirmed scientific theory. Let’s look more deeply into this matter. The result will be an appreciation of consciousness’s profound weirdness.
IT IS A SIMPLE, BRUTE FACT that early-twenty-first-century science doesn’t know what neural processes consciousness is associated with (I’ll use “science” as a catch-all term to denote neuroscience and psychology and any other science one might care to deploy in the quest to understand consciousness, like quantum physics). The list of candidates proposed by scientists to explain consciousness is embarrassingly large. Here are some of them: attention, autobiographical memory, being awake, body-based perspectivalness, episodic memory, executive processing, feedback, feature integration, 40 Hz neural oscillations in human brains, high-level encoding, intentionality (as in intending to do something), intentionality (as in a thought’s content or its being about something in the world), metaprocessing, mind-based perspectivalness, neural competition, quantum effects in the microtubules of neurons, recursivity, reflective self-awareness, reportability, salience, and sense of self.
It is rare in any science of any type to have such a long list of hypothesized causes of some phenomenon. But the truly alarming thing about this list is not how long it is, but how diverse it is. It contains proposed hypotheses that are themselves deeply mysterious and so of no help at all (for example, sense of self), along with actual measurable neural processes (for example, 40 Hz pulses), along with actual measurable psychological processes (for example, episodic memory), along with some very eyebrow-raising physics (for example, quantum effects of the microtubules of the neurons in one’s brain)—all of which have so far produced no viable theory of consciousness.
But, in truth, everything on this list is a waste of time: science cannot touch consciousness. The list of things about consciousness that we cannot explain is not just huge—it is absolutely every single thing associated with consciousness. Scientists cannot explain at all how the five senses work to produce sounds of ocean waves or visions of the color indigo. Scientists know how to numb neurons for light surgery and, for more serious surgery, the anesthesiologist can put you to “sleep.” But no scientist anywhere knows why the gases and chemicals remove consciousness, either in part or more completely. We don’t even know why aspirin makes headaches go away. We know it works, and we know in some detail what aspirin does, neurochemically. But we don’t know why those neurochemical reactions are or cause the cessation of pain. Neurochemical knowledge is essentially pointless when it comes to explaining experience. No one knows why whisky or wine or any other such chemicals have the effect that they have. No one knows why LSD or marijuana do what they do. No one knows why love or orgasms feel the way they do. No one knows how neural processes produce sadness or grief. No scientist knows why Prozac or Xanax or any of the drugs used to treat any mental illness or condition do what they do. And no one knows what neural processes produce joy or ecstatic spiritual knowledge. Again, much of the neural chemistry of all of these is known; scientists know a great deal about how these chemicals work on the neurons in the brain and the rest of the body. But all this neurochemistry is useless for explaining why these chemicals have the effect on conscious experience that they have. These experiences are what matter, and we cannot explain them.
It is not that scientists don’t know vast quantities of stuff about neurons, neural architectures, neural processes, brains, psychological processes, psychological states, and so on. They certainly do: attend a neuroscience conference or read an issue of Science and you’ll quickly find out how much they know; it’s immense and very technical. For example, they know much about the neural processes underlying vision, hearing, smelling, tasting, and touch. But they are clueless about why those neurons, those neural processes, produce the conscious visual experiences that they do. No one knows why neural firings produce pain, nor why a headache hurts differently from a punch in the face. The anesthesiologist knows tons about what the gases and chemicals she’s putting into you do to your neurons and brain. But she has no clue at all about why doing those things to your brain and neurons makes you lose consciousness to such an extent that a heart surgeon can open up your chest while you lie perfectly still. No clue. And I am not exaggerating one whit; I’m being literal and precise.
Contrast this with cars. It would be strange indeed if mechanics knew a ton about how cars are put together, but didn’t know why turning the key in the ignition started cars. They in fact know in detail why turning the ignition key works. And some know what’s going on down to the electron level. That’s good if you own a car, especially one that’s not starting. More deeply, it shows what human knowledge can accomplish. That we can know something so thoroughly is one of the very best things about humans. But human knowledge can’t touch consciousness.3
You are probably dubious. Surely, you say, if science can’t explain consciousness now, it will eventually, perhaps even soon. Perhaps you even think that nothing is intrinsically beyond science’s purview. I’ve had many a discussion with cognitive scientists and neuropsychologists who of course agree that science cannot now explain consciousness, but who think it will before the next decade (next fifty years, next century) is over. (Different researchers provide different lengths of time.) I call this the “just wait until next year” claim; it should be called the “just wait until next year” blind faith. Just wait until next year. . . . we can wait until the next millennium for all the good it will do.
Here are two intuition exercises (similar to thought experiments) to show that consciousness is beyond explaining. As luck would have it, there are two very popular movies that make this exercise fun.
The first will exercise what is called your Cartesian intuition (named after René Descartes). This is the intuition that your conscious experiences could be just what they are no matter what the world is really like. I am confident that you have this intuition; it is well known that most people have it. The famous movie The Matrix is founded on exactly this intuition, and in fact is an extended, multimillion-dollar celebration of it.
In the movie, the entire human population is really living out pathetic “lives” encased in billions and billions of underground coffins filled with some sort of life-sustaining goo. Each hairless body in each coffin is attached to a world-sized computer that directly gives all of our brains the experience of living our lives on the real surface of Earth sometime around the turn of the second millennium ce (common era). The “reality” we all know and love—life on planet Earth—is entirely simulated: no one has even once actually used his or her eyes, or any other sense organ, to sense anything. We are all in one large, collective computer dream. In the twenty-first century, we could call this the Matrix intuition, but the famous philosopher René Descartes (1596–1650) beat the Wachowskis to it (by a few years). Descartes imagined an evil demon manipulating our senses (note: not our sense organs [for these are physical things we believe we have] but the actual conscious experiences associated with our five senses) in various ways in order to deceive us into believing that we have human bodies and are walking around doing things. In fact, we have no bodies at all. Reality as we conceive it is an illusion. In his Meditations on First Philosophy, Descartes describes this demon thusly: “as clever and deceitful as he is powerful, . . . [he] has directed his entire effort to misleading me.” Why would such a demon spend his life (or eternity) doing such a thing? Why because he is an evil demon. (In The Matrix, the builders of the matrix have a much better reason, it turns out.)
(There’s a tiny wrinkle we need to discuss. I claimed that Descartes said the evil demon manipulated our senses, not our sense organs. He was careful about this because, in truth, we may not have sense organs, or at least not the ones we think we have, if the evil demon is fooling us. A good way to imagine this is to think that we are just nonphysical, disembodied spirits—ghosts—being fooled by the demon. But in The Matrix, humans in fact have human bodies; it is our brains and sensorimotor systems that are being stimulated in the right way to fool us into thinking we are reading a book on religion and scientific mysteries. The Matrix is a lot more believable nowadays, a lot more easy to imagine, than Descartes’s disembodied/evil demon version [in fact living in a matrix may well be in our future, if AI and research into virtual realities unfold in a certain way].4 We have trouble with the disembodied version because we tend to think of the evil demon as using some sort of physical apparatus on us, and we find it hard to imagine causally affecting something nonphysical [our ghost selves or our spirits] with something physical. But perhaps the evil demon isn’t using something physical on us. He/she/it is using something nonphysical, perhaps magic. If that works for you, great. I invite you to use either The Matrix or Descartes’s evil demon version to pump up your Cartesian intuition. Either will work fine.)
The Cartesian intuition is easily and naturally held simply because we all dream at night. You are hang gliding in the beautiful Himalayas . . . but no, you are asleep in your bed, dreaming. Perhaps you have never hang glided in Himalayas, or perhaps you’ve never hang glided anywhere. Perhaps you don’t intend to; perhaps you don’t even want to. But there you are anyway—soaring between Chomolungma and Nuptse. But dreams aren’t “real.” So we all know that we can easily experience something quite robustly even though we aren’t “really” doing what it takes to actually have that experience (for example, we aren’t actually hang gliding in the Himalayas).
The Cartesian intuition, then, sunders the connection between the world and your experiences. Or at least it shows that the connection between your experiences and the world is tenuous. You can experience anything, yet be doing basically nothing. You are not now actually reading this sentence, using physical paper and ink, and light, and your eyes. . . . No, you are encased somewhere in some vat of goo, being duped into thinking that you are actually reading this sentence. Or perhaps you are dreaming that you are reading this. Perhaps you are a computer, dreaming that you are a human reading about mysterious spiritual journeys.
Of course, it is next to impossible to believe for any length of time that you are actually in a vat of goo in the matrix (or asleep in your bed or whatever). You think you are in the world, living your life, reading this sentence. This is a perfectly rational belief—if you didn’t have it, you’d probably be in a hospital somewhere being treated for some sort of mental illness. What the Cartesian intuition shows is that, though perfectly rational, your belief that you are living your life in the world is in fact a leap of faith—the connection between your experiences and the causes of those experiences is nowhere near as tight as you think. In fact, as far as you know, there’s no connection at all.
Though the Cartesian intuition discombobulates matters considerably, we can discombobulate matters further. To completely sunder the connection between consciousness and the world, we can assume that life is really as we experience it, that there is no matrix, and that our brains are filled with neurons . . . just as it seems. Now, we will work to imagine neurons doing their neuronal thing with no consciousness at all. We will clearly and cleanly imagine a being of some sort living a robust life, traveling hither and yon, doing this and that, with neurons abuzz, yet with no consciousness at all. This is where the second intuition exercise comes in. And the second movie.
Philosophers have a technical term for creatures that have bodies—neurons, brains, and sense organs—but experience nothing: zombies. Zombies are very important in philosophical discussions of consciousness, because they seem to show that one can be fully embodied and alive and yet completely inert on the inside. (I don’t mean inside their skulls: zombies’ neurons work perfectly well. By “inert on the inside” I mean zombies lack a self that is the locus of conscious experience. Note: philosophy zombies are very different from monster zombies portrayed in popular fiction.) To use Nagel’s phrase: there is nothing it is like to be a zombie—or, using a slight variation, being a zombie is just like being a doorknob: experience-less.
Well, how easy is it to imagine zombies? Can one legitimately imagine them? The zombie intuition, the intuition that zombies could exist here and now on planet Earth, is not as easy to come by as the Cartesian intuition. Fortunately, Hollywood has helped out here, too. In Pirates of the Caribbean, Captain Barbossa and his mutinous band of miscreants are cursed men. After stealing quite a bit of sacred Aztec gold (which had had a curse placed on it by the Aztec gods—understandable given Cortez’s behavior), Barbossa and his men discover that though they can feel profound, gnawing hunger, choking thirst, and consuming lust, they cannot in any way satisfy their cravings. In fact, they can’t feel or sense much of anything, not the wind on their faces, nor the smell of the salt air. They can only feel the three desires just discussed and pain. Come to find out, though by day they seem to have bodies and skin and organs, in the light of the full moon, one can see what they really are: corroding skeletons.
Barbossa is not a zombie . . . not quite. He can still feel some things (and of course, he doesn’t have neurons, at least by the light of the full moon—we will let that slip, however). But we can use Barbossa to construct a zombie, a zombiefied Barbossa. Here goes. . . .
Let us first assume that Barbossa can feel nothing at all. He forces Ms. Turner to take off her dress before he forces her to walk the plank (never fear, Pirates is rated PG-13, so Ms. Turner remains fully clothed at all times, in a totally modest undergarment of some sort). He leeringly says of the dress, “Mmmm . . . still warm . . .” When he does so, he (Barbossa) cannot actually feel the warmth still in the dress. So how can he correctly say, “Still warm”? Well, his body can still process the information that the dress is warm, much like a thermometer could. Thermometers aren’t conscious (as far as we know), yet they accurately register temperature. Barbossa’s bodily heat sensors work like that.
Though one can carry on a conversation with Captain Barbossa—he seems to see and hear you and responds accordingly—we can easily assume that he doesn’t have any visual or auditory experiences at all. His sense organs (such as they are) process the information that exists in the relevant light and sound, and Barbossa’s body (mouth, tongue, and vocal cords, for example) produces the right output . . . but during all of this, Barbossa experiences nothing. His ears pick up sounds like microphones, his eyes pick up light like cameras. Mechanisms process this information, but no experiences ever happen. The same is true, let us suppose, of smell and taste. Both of these work like chemical analyzers, checking for various chemicals and outputting the correct response, but no experience accompanies this information processing and this response. (In the movie, Barbossa can experience taste: referring to himself he says everything tastes of ashes. But let’s suppose that his taste experiences have also become zombiefied; he can taste nothing.)
So, all five of his senses work just like some unconscious machine or device: thermometers (and various pressure-sensitive devices) for touch, microphones for hearing, cameras for vision, and chemical analyzers for smell and taste. On the inside, Barbossa’s experiences are only hunger, thirst, lust, and pain. He doesn’t even experience darkness.
Now, if we remove his feelings of suffering—the hunger, thirst, lust, and pain—we get our zombie (in the movie, some of Barbossa’s pirates seem to experience pain when stabbed or hit). So, Captain Barbossa can be rendered as a philosophy zombie: a perfectly physical being who behaves in all the right ways all the time, just like you and me, but who is utterly inert on the inside. There is nothing it is like to be our zombiefied Barbossa. In fact, in many ways, it is far worse to be the actual Barbossa than the zombiefied Barbossa, since the actual pirate suffers hunger, thirst, and lust. (C3PO and R2D2 [from Star Wars] and Commander Data [from Star Trek: The Next Generation] might possibly also help you imagine zombies, but this is unlikely, since what really happened in those shows was that we all became convinced that robots [androids] could in fact be conscious. This intuition is just the opposite of the zombie intuition.)
Still, you might be reluctant to grant that Barbossa, or any zombie, could behave perfectly correctly and yet be unconscious. I have an ace up my sleeve that will answer this reluctance.
For all you know, your spouse, partner, best friend, dog, cat, and so on could be a zombie. Try this. Pick out someone near you now (actually, it helps if you pick out a stranger), and imagine that that person is an experience-less zombie, an unfeeling, conscious-less android of some sort, capable of all the right behavior but utterly inert on the inside. Imagine that there is nothing it is like to be the person you picked out. You know they behave appropriately, but you don’t know that they are conscious. The same is true of you and me: your consciousness is unavailable to me, and mine is unavailable to you. Consciousness is entirely and forever internal and personal. Your heart is internal, but I can make it external—heart surgeons do this routinely. Consciousness is forever internal to one person, particular to that person and that person only. This isolation gives rise to what philosophers call the problem of other minds: a person with a conscious mind, me for example, doesn’t know for sure that there are other conscious minds in the world; there might only be zombies. The problem of other minds arises when we come to the conclusion that behavior isn’t sufficient for knowledge that someone else—anyone else—is conscious. (Actually, the problem is more acute than doubting the consciousness of others, which is bad enough; others may also not have any thoughts at all, conscious or otherwise, for example, the thought “oh, it’s time to walk the dogs.” Everyone else may be very sophisticated puppets of some sort. We will skip this more virulent version of the zombie intuition.) We see that consciousness (and the mind in general) is very peculiar. Its hallmark is utter and eternal isolation.
So, given everything you know about someone else—which is all derived from his or her behavior—that person might not have any consciousness at all. The person you are closest to, the person you love the most, might be a zombie. Your consciousness might be the only consciousness in the entire universe. Or perhaps my consciousness is the only consciousness in the entire universe. I can only be certain of my consciousness. From my perspective, which is the only one I am prepared to swear to, all the rest of the animals on planet Earth might be zombies. This isn’t solipsism (which is the view that only I exist), but it is disturbingly close, probably too close.
I want to stress that for all you know, your own true love is a zombie. Note that it is impossible to prove that your spouse, say, is a zombie, just as it is impossible to prove that he or she is not a zombie. Imagine if someone asserted that you were a zombie. How would you refute them? Drop a bowling ball on your foot and then cry out in pain? Zombies can mimic this behavior exactly. Eat some yummy chocolate ice cream and say, “Yum?” Zombies can do that, too. State that, upon reading Excellent Beauty, you find consciousness to be mysterious? Again, easily replicable by a garden-variety zombie.
Zombies, then, demonstrate that physical processes (for example, neural ones) can go on without consciousness existing at all. Zombies finish what the Cartesian intuition started: consciousness is now completely sundered from the physical. Your conscious mind is now shown to be independent of the world it inhabits and even of the very neural processes it seems most closely associated with. I’m not saying that you can believe this conclusion on a day-to-day basis. You will find it impossible to believe for any length of time that this book you are now reading isn’t really out there, in your hands, causing your book experiences. But I am saying that for brief moments of time, you can see that your consciousness could well be utterly independent of any world you think is out there. In fact, there need be no world out there at all.
Now, it is easy to see that consciousness is mysterious indeed. But we aren’t done with consciousness yet.
I said that human knowledge can’t touch consciousness. I meant that external, third-person knowledge can’t touch consciousness. In one very important sense, human knowledge not only can touch consciousness, human knowledge is constituted by it. Remember that your conscious experiences are forever beyond me. But they are you. You know nothing as securely as your own conscious states, your own experiences. Your consciousness, in fact, is the most important thing about you, for if you weren’t conscious, if you were a zombie, then there is an important and real sense in which you wouldn’t exist. That your physical body exists is the second most important fact about you. As I mentioned, your consciousness—the fact that there is something it is like to be you—is what makes your life worth living, or what makes you want to die right now, or quit reading and go get a sandwich. In fact, in some way that—big surprise!—no one understands, consciousness gives rise to or is somehow intimately associated with your ability to use language to mean and communicate things, to have knowledge, and to be moral. So, of course, all these are going to remain at least partially mysterious since consciousness is. (By the way, their deep connection to consciousness explains why these three remain philosophical problems and not the problems of some science.) And most importantly, conscious is crucially tied to your self—that inner nexus of perception and control which is you. The self is another locus of mystery. In fact, in these physicalist times, many philosophers deny that there is any such thing as the self.5 That’s how utterly strange the self and (its?) consciousness are: one can be a conscious self and yet in (apparently) good conscience deny that one is a conscious self.
Infinity and Beyond
Approximately how many infinities would you say there are? Let’s make things as simple as possible: let’s restrict our discussion of infinity to numbers, to mathematics. I know this might put some people off, but at least with numbers, matters are quite precise. It is this very precision however, that makes things here startling, perhaps even unsettling.
1, 2, 3, 4, . . . these are the counting numbers (let’s deal with 0 later). We all use them every day: to tell time, to handle money, to solve important problems in our lives. We all know that there are an infinite number of them—they go on forever (you can always add 1 to any counting number to get the next biggest number). The counting numbers are made up of even numbers (numbers cleanly divisible by 2) and odd numbers. There are an infinite number of even numbers and an infinite number of odd numbers. So we get:
2 4 6 8 . . .
1 3 5 7. . . .
We can put all these sequences in alignment:
1 2 3 4 . . .
2 4 6 8 . . .
1 3 5 7. . . .
One can see by this alignment that for every counting number, there is an even number and the same is true for the odds. And we can see that for every even (and odd) number, there is a counting number. We can conclude that there are exactly as many counting numbers as even numbers and odd numbers. Yet, the even numbers are a strict subset of the counting numbers, as are the odd numbers. By this, I mean that the even numbers are only a part of the counting numbers (same with the odds); yet, though the evens are only a part of the counting numbers, and though a part is always smaller than the whole, there are exactly as many of both of them. How is that possible? Well, it has to do with infinity. Ahhh . . . but which infinity?
One of the things counting numbers count is how many members sets have. In fact, you can think of counting numbers as only counting sets of things. There are (probably) 10 fingers on your two hands. We can think of your hands as making up a set of hands with 2 members. We can think of your fingers as making up a set of 10 fingers. If you count the number of people you work with or the number of red cars in a city parking lot, you are counting the number of members of sets of people and cars, respectively. Mathematicians call such numbers cardinal numbers. Cardinal numbers tell us how many things there are. The cardinal number (or just cardinality) of your set of fingers is 10, of your hands is 2. The cardinal number of the set of your heads is 1—that is, you have 1 head (unless you’re Zaphod Beeblebrox, of course).
Mathematicians have a number that they use to count all the counting numbers . . . all infinity of them. This number is ℵ0 (pronounced “aleph naught” or “aleph null”—the subscript 0 [null] is going to be important). Where you might write ∞, mathematicians, being more precise, will write ℵ0. As we will see, ∞ is ambiguous. Mathematicians therefore say that there are ℵ0 counting numbers, or, slightly more precisely, that the cardinality of the set of counting numbers is ℵ0. The cardinality of the evens and odds is also ℵ0. So, the size of the (set of) counting numbers and the size of the (sets of) evens and odds are the same—ℵ0—even though the evens are only a part of the counting numbers, and even though, usually, a part is smaller than the whole.
“Fine,” you say, “sometimes parts are the same size as the whole; infinity is bit strange. . . .”
Well, the counting numbers, as you know, aren’t all the numbers. We frequently find that there are 0 things in some set. The size of the set of gold bars in your basement is (probably) 0. We are missing 0. If you cut an apple in half, you have 2 halves. A dime is.1 of a US dollar. We are missing the fractions. If you are in debt to, say, your bank to the tune of $10,000 for your recently purchased (used) car, then you have −$10,000. If you have $1 in your pocket, and buy $2 worth of candy, then you are −$1 in debt (hopefully you are with a friend with +$1). We’re missing the negative numbers, too. And we are missing the negative fractions.
Adding 0 is easy. There is only one 0. We can consider it to be even and odd. So, adding 0 won’t change anything.
0 2 4 6 8 . . .
0 1 3 5 7. . . .
There are still ℵ0 numbers in each sequence.
How many fractions are there? Infinity. How many negative numbers (including the negative fractions)? Infinity. Yes, but are there ℵ0 of them all or more than ℵ0? It seems as if there must be more than ℵ0 of them.
If you put into one set all the counting numbers, zero, the negative versions of the counting numbers, and all the positive and negative fractions, you get a set of numbers that mathematicians call the rationals (they are called this because every number in the set can be represented by a ratio of integers, that is, counting numbers or their negatives: for example, 2/3, 2/1, 4/2, −78/193, and so on). Okay, so the question is, how many rationals are there? Surprisingly, there are ℵ0 of them. This is strange because intuitively the rationals ought to be far larger than just the counting numbers (which lack all the negative numbers and all the fractions). Here’s the proof.6
Consider the array in figure 10.1 below:
This array contains the entire set of rationals (of course only a few are actually written down). First, there’s zero. Then the first row has all the positive and negative whole numbers, then the second row has all the positive and negative fractions with 2 as the denominator, the third row has all the positive and negative fractions with 3 as the denominator, and so on. The first column has only 1s in the numerator, the second, only −1s, the third only 2s, and the fourth only −2s, and so on. Given any rational number whatsoever, we can find it in this array (of course, this might take a while . . . perhaps the life of the universe up till now, but in principle we can find any number in the array). The array, by the way, contains some redundancy. For example, 1 is represented by 1, 2/2, 3/3, 4/4 . . . 145/145 . . . and so forth; there are other redundancies, too, such as 2/3 and 4/6. This is okay, and won’t affect anything since, as mentioned in figure 10.1, the redundancies are avoided.
So far so good. Now, we know that there are ℵ0 counting numbers. So if we can place each counting number starting with 1 with exactly one rational number, and if we can guarantee that we will cover all the rationals this way, then we will know that there are ℵ0 rationals. The arrows in figure 10.1 show you how to do just this. The arrows sequentially count the rationals (skipping over the redundancies), giving us:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 . . . |
| 0 | 1 | 1/2 | −1 | 2 | −1/2 | 1/3 | 1/4 . . . |
The ℵ0 counting numbers are the top row, and the rationals make up the bottom row. As can be seen by how the arrows in figure 10.1 progress (to infinity), each rational number will be accounted for: each rational will be counted; none will be missed. Hence, there are ℵ0 rationals.
FIGURE 10.1. This array shows the beginning of the entire set of rational numbers. The arrows show how to move through the array in order to pair each rational number with exactly one counting number and vice versa, thus counting each rational number, and hence showing that there are exactly as many rational numbers as counting numbers. Repeats are excluded, however, so once, say, 2/3 has been counted, 4/6 is excluded.
So we have now established that even though, intuitively, there seem to be fewer evens than counting numbers (because a part is less than a whole), there are in fact exactly as many evens as counting numbers, namely, ℵ0 (same with the odds). And we’ve established that adding all the negative numbers and all the fractions to the counting numbers also leaves us with exactly the same quantity of numbers, namely, ℵ0. So, though intuitively adding the infinity of fractions and the infinity of the negatives to the infinity of counting numbers would make a set with more numbers (because adding always makes more), doing so here changes nothing. There are exactly as many rational numbers as there are counting numbers. Or as the mathematicians say, the cardinalities of the counting numbers and the rationals are equal, namely, ℵ0.
“OK, infinity is more than a bit strange . . .”
We’re not done yet: We still don’t have all the numbers we use routinely. We are missing the irrationals. When represented in their decimal form, all rational numbers either terminate in a finite string of numbers or fall into a repeating pattern. For example, 1/2 is.5, 1/3 is.33333. . . . . Irrationals are not like this. They are numbers whose decimal representation never terminates and never repeats. Numbers like π and 3√7 are good examples. The number π is famous because you need it to calculate the area or circumference of a circle (it is also one of the most important numbers in all of science). π is the ratio of a circle’s circumference to its diameter (but π is not a rational number because it is not a ratio of two integers).
Ok, once you add all the irrationals to the rationals, you get a set called the real numbers. And guess what? You no longer have only ℵ0 numbers. Here’s the proof.
Proof That the Cardinality of the Real Numbers Exceeds ℵ0
This proof was also first developed by Georg Cantor in 1891. The version presented here is derived from both Wallace’s and Dunham’s books. The proof is called Cantor’s Diagonal Proof, for reasons that we will see. It is a proof by contradiction. This means that we will assume that all the real numbers (aka the reals) can be put into a one-to-one correspondence with the counting numbers, just like the rationals. This is equivalent to assuming that the cardinality of the set of reals is ℵ0. Then we will produce a new number outside of this correspondence, thereby showing that our initial assumption about the size of the reals was false: the cardinality of the reals must be greater than ℵ0.
First, assume that the reals can be placed in a one-to-one correspondence with the counting numbers, as exhibited in table 1.
This table shows the counting numbers paired with fractions, negative numbers, irrationals, and rationals. If this correspondence worked, then every single real number (every single rational and irrational number) on the right would be paired with a counting number on the left, and vice versa.
However, we can construct a new number, b, that is different from every number in table 1, and, hence, that is not in the table. First, we are going to ignore all the digits to the left of the decimal point. If we can construct a number that differs from every number in table 1 in just the digits to the right of the decimal point, then we have shown that b is not in table 1, which is what we really want to do. So, assume that b = .b1 b2 b3 b4 b5. . . . We build b as follows:
For b1, if the first postdecimal digit of the number paired with 1 is not 5, then pick 5; if the first postdecimal digit of the number paired with 1 is 5, then pick 4;
For b2, if the second postdecimal digit of the number paired with 2 is not 5, then pick 5; otherwise, pick 4;
For b3, if the third postdecimal digit of the number paired with 3 is not 5, then pick 5; otherwise, pick 4;
. . . and so on . . .
In general, for bn, if the nth postdecimal digit of the real number paired with counting number n is not 5, pick 5; otherwise, pick 4.
What we’re doing is moving down the diagonal of the list of real numbers in table 1, picking 5 for each bi each time the number on the diagonal is not 5, and picking 4 if the number is 5. Look at table 2, which is just table 1 with some highlights, and notice the underlined digits.
Our new number, b, is.54555. . . . As can be seen by inspection, b is not in table 1 (again, we are ignoring what comes before the decimal point, if anything does). The number b is not the first real number (the one paired with 1) because b differs from that number in its first postdecimal position. b is not the second real number (the one paired with 2) because b differs from that number in its second postdecimal position. b is not the third real number (the one paired with 3) because b differs from that number in its third postdecimal position. . . . and so on . . . In general, b differs from every real number, R, in R’s nth position, where R is paired with the nth counting number.
By following this construction procedure, our new number, b, is guaranteed to differ from every number in table 2 in at least the position shown by the underlined digit. This, in turn, means that b is not in table 2 (remember, table 2 is just table 1 with certain digits highlighted). But our initial assumption was that the reals could be paired with the counting numbers, and so b should be in the table. But it’s not. Note that adding b to table 2 (suppose we pair it with number 1 and move all the others down one) will not help, for we can just repeat our recipe for constructing yet another new b. So, the assumption that the reals are pairable with the counting numbers must be false. Hence, the cardinality of the reals is greater than ℵ0—there are more reals than there are counting numbers and hence more reals than there are rational numbers. This completes Cantor’s Diagonal Proof.
Well, if we don’t have ℵ0 numbers, then how many do we have? An infinity, of course. Only now, we have an infinity that is “bigger” than ℵ0. Mathematicians call this infinity ℵ1, and ℵ1 is strictly bigger than ℵ0.7 So, there are ℵ1 rationals and irrationals together (that is, there are ℵ1 real numbers). Since there are only ℵ0 rationals, there must be ℵ1 irrationals. So the irrationals, while infinite (∞), are vastly more infinite than the rationals, which are themselves infinite. Infinity, it appears, comes in sizes.
Now, let’s consider ℵ0 and ℵ1. Are these the only two infinities? Take a wild guess. There are an infinite number of infinities. This deep and puzzling idea was also discovered by Cantor in the late 1800s. All of these infinities exist in what is called the transfinite realm: ℵ0 ℵ1 ℵ2 ℵ3 ℵ4 ℵ5 ℵ6 ℵ7 ℵ8 ℵ9. . . . Consider ℵ9. It counts an infinity that is much much larger (if we can use “larger” here) than ℵ0. How can this be? And there are an infinite number of transfinite infinities far beyond mere ℵ9.
There’s more. I listed the first ten transfinite cardinals as if they could be paired off with the counting numbers. So, it might appear that there are only ℵ0 transfinite numbers (also called transfinite cardinals), that is, one might think that the cardinality of the counting numbers and the cardinality of the transfinite cardinals are the same. But this is an illusion of listing the first of the transfinite cardinals. There are so many transfinite cardinals that mathematicians say that they cannot even be put into a set, no matter how large the set, and sets can be very very large. There are so many transfinite cardinals that there is no cardinal number to count them. Yet there are an infinite number of them. We are left with the notion that some infinities are bigger than others, bigger by sizes that are just barely conceivable.
So infinity is not simple. It upsets matters considerably.8
The Rarity of the Commonplace
How rare are you? How rare are the things you know and love best? . . . A moment’s reflection will reveal that you are not just rare, but unique. Your particular combination of everything from your genes and gene errors to your experiences is not duplicated anywhere in the universe. In fact, they are not now duplicatable, since much of what you experienced is gone for good. But abstract away from your details. How rare is what you are made up of—matter and energy? They seem to be the most common thing there is.
Rare things are interesting, in part simply because they are rare—seldom seen. A royal flush in poker is a winning hand because it is so rare. A dust devil blowing across the Martian landscape is rare (one was photographed by the Phoenix Lander during the early Martian Fall of 2008). But when something that is utterly common is shown to be actually profoundly rare, that is somewhat strange. That’s what we’ll do here: the common will be shown to be rare. This sort of rarity seems to violate the very definitions of “rare” and “common.” So, the kind of rarity we are interested in here is not just the rarity of the seldom seen. Rather, the rarity we will examine violates intuitive, fundamental principles governing the very notion of rarity.
The rational numbers we discussed above are so rare as to be almost nonexistent. Let’s look at this (the discussion to follow can be seen as a consequence of the fact that ℵ0 is smaller than ℵ1). The rationals, as we know, are infinite in number; there are ℵ0 of them. But unlike the counting numbers, between any two rational numbers, there is a third (simply subtract the smaller of the two numbers from the larger, divide by 2, and add that quotient to the smaller of the two rationals). The counting numbers lack this property. There is no counting number between 2 and 3. But there are rational numbers between the rationals 2 and 3: 2-1/2 for example. In fact, between any two rational numbers, no matter how close together, there is an infinite number of other rational numbers. This property of the rationals is called density: the rationals, but not the counting numbers, are dense. Which infinity counts the cardinality of the rationals between any two other rationals, you now ask? Good question. Answer: ℵ0. So between 1 and 2, or between 2 and 4, or between 2 and 70 quadrillion, there are exactly ℵ0 rational numbers. And there is a total of ℵ0 rationals altogether. The notion of density makes the fact that there are exactly as many rationals as counting numbers even more startling.
Yet we know that there are far more irrationals than rationals. In fact, there are so many irrationals that ℵ1 of them cannot even be given names of any sort (and I didn’t just name them by using the phrase “cannot even be given names of any sort” because that names the collection of them, not any particular one).
Now consider what is called the Real Line, the familiar line from high school math classes. This line is made up entirely of the rationals together with the irrationals. Any number from this line is called a real number. So a real number is either a rational or an irrational. There are infinitely many rationals and they are infinitely dense, so how much of the Real Line would you expect them to take up? How much space on the Real Line do the rationals occupy? . . . An infinitesimally small amount—basically none.
Imagine that you have the Real Line in front of you. Not a line drawn on a chalk board or on the sidewalk or on a computer screen, but the Real Line—the actual thing itself. And imagine that you have a dart whose point is sharper than infinitesimally sharp—it can hit exactly one point on the line (physically impossible, but conceptually okay). You throw the dart at the line (suppose you are an expert marksman with darts). Mathematicians have proved that the probability of you hitting a rational number with your very special dart is . . . 0 percent. Essentially, the Real Line contains no rational numbers at all. They are so rare as to be vanishingly nonexistent. The Real Line is really made up of irrational numbers. But further, and continuing with the strangeness, the Real Line is in fact mostly made up of exotic irrationals like π, not “garden-variety” irrationals like √2 or 3√7. Mathematicians call exotic irrationals like π the transcendentals. Yet, the numbers you know best are the rational numbers. The numbers that we encounter most of the time, by far, are the rational numbers, the very numbers that basically don’t exist. Searching randomly in the vastness of all the numbers, rational and irrational alike, you will likely never find a single rational number. They are rarer than perfect diamonds, than Albert Einsteins, than Mozarts, than lightning striking the same place over and over, every second for a thousand years . . . but they are not as rare as you.
Let’s abstract away from your particular details. Let’s just consider the stuff you are made of—your mass (matter) and energy. At one time in the not too distant past, it was thought that everything in the universe was made of mass or energy (they are interchangeable, via Einstein’s famous equation, E = mc2, which says that the energy, E, of a mass, m, is that mass times the speed of light, c, squared). But no more. Now cosmologists hypothesize that the universe contains, beside mass and energy, dark matter and dark energy. Cosmologists explain that these terms don’t mean that the matter and energy are evil or somehow foreboding. Rather, the word “dark” in the case of dark matter means that such “matter” doesn’t emit radiation in the electromagnetic spectrum, which is very unusual, and in the case of dark energy, the term “dark” is used to flag the fact that this energy works against gravity, pushing the universe apart. But also, in both cases, the term “dark” is used to mean “unknown.” “Dark” functions here in the same way the word “incognita” worked when cartographers of old labeled some geographical area “terra incognita.” (Still, if the cosmologists had really wanted to, they could have called dark matter “nonradiating matter” or “unknown matter” and dark energy “negative pressure energy” or “unknown energy.” But they didn’t. Makes you wonder why; makes you wonder if there’s something they know but won’t, or can’t, tell us.)
Though we don’t know what it is, dark matter is thought to exist because our Milky Way galaxy, for example, is rotating so fast that it would fly apart unless there were a lot of extra matter within it holding it together. We just can’t see, or seem to find, this extra matter. We infer the existence of dark matter solely from its alleged gravitational effects. Dark matter is now speculated to make up somewhere between 22 and 30 percent of the mass of the universe. Dark energy (which was only discovered in the very late twentieth century) is thought to make up about 66 to 74 percent of the universe. And the remaining 4 percent of the stuff of the universe? That stuff is ordinary matter and ordinary energy. So, all the stuff you know and love (and hate and are indifferent to) and all the energy that makes it possible—the pies, cakes, your favorite humans, family members, the stars at night, light, the sun, Earth, roller derby, roller coasters, mountain climbing, sex, work, housecleaning—makes up about 4 percent of the universe. This means that we only understand about 4 percent of the universe. Our physics explains that 4 percent, but unfortunately says little about the remaining 96 percent.
So, not only are you unique, but the general stuff of which you are made, and that powers you, your muscles, and your cells, is rare indeed in this vast universe of darkness. The rest of the universe is made up of . . . well, God knows what, . . . perhaps not even he/she/it knows. . . . This would explain a lot, especially if the cosmologists know something we don’t, and if, though sworn to secrecy, the most decent among them tried to warn us, with the only clue they could let slip out . . . about something very very dark.
So, cosmology, the study of the very large (from suns to galaxies to galactic clusters to superclusters and beyond), introduces rather strange physics. The other end of physics, the very small, also has its share of strangenesses and mysteries. I speak of course of quantum mechanics. The issue of what’s rare and what’s common takes a bizarre twist here. (I will skip all of the standard quantum mechanical weirdnesses such as matter being both particles and waves at the same time and quantum-induced shadow universes because they are well discussed in many other places.)9
One of the most common things in the universe is causation. I mean ordinary, everyday causation. Jones throws the bowling ball at the bowling pins, hits them, and the pins fall over. Smith sticks her car key in the ignition, turns it, and the car starts. Bartleby has a cold, he coughs on you, and you catch his cold. John’s lung cancer killed him. The sun comes up and warms the farmland and its crops, then the rains come, and the crops get bigger. Jiang Lee and Sharise have sexual intercourse, and Sharise gets pregnant. And on and on and on. Causation is what makes us victors and victims in this world. We make stuff happen. And other things make stuff happen to us. Causation is so common that we almost never think about it. We usually only think about it when something goes wrong. The bowling pins didn’t fall over. The car won’t start. Sharise doesn’t seem to be able to get pregnant. John never smoked. What would cause someone to spend months and months learning to fly a plane only to hijack one and fly it into a large building? What could cause them to think that was a good idea? And so on. . . . Causation is what makes it possible to be in this universe, and what makes it possible for there to be a universe to be in.
The quantum world, however, is completely bereft of causation. The very essence of quantum mechanics is pure probability. In the quantum world everything happens with some probability and that is all there is to say about it. Nothing causes anything at all. Stuff just happens. Sometimes the stuff that happens (an electron going a certain direction) is very probable, and sometimes it is very improbable (an electron passing through a certain barrier). But none of it is caused. So what seems to be the most common thing in the universe, the thing that makes it intelligible to us (namely, causation), simply disappears at the fundamental levels of the physical universe. In essence, causation emerges from the realm of the noncausal.
Physicists have partial explanations of some of the physical aspects of this emergence (why bowling pins fall over when hit by bowling balls), but their explanations can’t be extended to explain why a hurricane causes the US economy to stagger. Economists can explain why hurricanes cause the US economy to stagger, but they can’t tie their explanations to physics. In fact, there is no general theory at all of causation, physical or otherwise. We don’t know in general what causation is, or why it works the way that it does, or why it varies as much as it does. The situation is so bad that the philosophers are involved (never a good sign). David Hume (1711–1776), a Scottish philosopher, famously argued that there’s really no such thing as causation at all, at least we can’t know that there is such a thing. What we see when we see a moving bowling ball cause some bowling pins to fall over is really just that the bowling ball rolled up to the pins, the pins fell over, and the ball moved off in another direction. We see no causation at all. We just see one event after another—continuously, throughout our lives. The more one thinks about this, the more true this seems. Hume’s suggestion would certainly solve almost all of the problems of causation—by getting rid of it. But as usual with such philosophical solutions, its price is far too high: it is not even clear we could survive by thinking of our lives as an unconnected series of probabilistic events. More importantly, it is not clear that Hume’s view is true in the ordinary world.
Matters get stranger. Everything is made up of quanta, not only matter, but energy, too. Light is made up of quanta, as are you. So at root, we are just swarms of probabilities that all have properties like spin and charge and “color” (a property of quarks).10 So, which is it, causation or probabilities? The answer is that it depends on which level you are at. This isn’t much of an explanation, though, since we don’t know why there are the levels that there are or even why there are levels at all. The levels I speak of are well known, but their emergence is mysterious. Why is there such a thing as chemistry (chemical processes), which sits on top of particle physics? Why is there such a thing as biology (life), which sits on top of chemistry? Why is there such a thing as psychology, which sits on top of biology? These are just four of the larger strata, each with its own kind of causation. There are many more levels, and each breaks into more detailed substrata. Scientists cannot answer the level questions, except in piecemeal, here and there. Rather, these are profoundly deep philosophical questions. And I think we all know what that means.
So what’s rare and what isn’t is not simple, and certainly not obvious. But it is quite puzzling.
BUT IT ISN’T ONLY RARE THINGS that are mysterious, that invoke wonder. The furniture of the everyday can also invoke wonder and a sense of mystery, though this sort of mystery, being common, is harder to experience. This is because most of the stuff in our lives belongs to a category that we deem common and, more importantly, understood. But the category something belongs to—wife, husband, child, dog, table, fork, spoon, book, father-in-law, headache—is really just a thin veneer masking that thing’s inherent, intrinsic weirdness. Think of this book. But don’t think of it as a book, or even as an object; rather, think of it, as the philosophers say, as what it is in itself. Think of this book not in relation to you or to anyone or anything else, but solely in terms of it, itself. Or think of it, as the practitioners of Zen say, as what it was before it was a book.
Seeing the strange in the ordinary takes some practice, and perhaps not surprisingly, there are meditative techniques that exercise this capacity of ours. But it is not just the book that is ultimately mysterious. The book becomes mysterious when the user of the book is removed (which, interestingly, can only be done by the user him- or herself). Yet, an even deeper mystery obtains when the user is left in. Look at this book, or this page, or read these sentences. A visual process is occurring, involving light, along with a large number of other things, which ends with you being conscious of seeing this book, this page, of the meanings of these sentences. The book (or whatever your object of consciousness is) begins, or lies at one end of, this process of consciously seeing the book. The other end is you, the perceiver of the book and the one who is conscious of the book. This other end, this conscious self that is you, is one of the biggest mysteries in all of science, as we’ve seen. In fact, as we saw above, your conscious self is—if anything is—the central mystery of the cosmos.
WE HAVE NOW EXAMINED three things that are mysterious indeed: consciousness, infinity, and rareness and abundance. There are many many more (see the appendix to this chapter). All of these point to the same thing: you and I and the universe we live in are quite strange. And even though we live in the twenty-first century, a century steeped in exciting and startlingly successful science, a century in which, daily, mysteries are transformed into common knowledge, we see that some mysteries, deeper than science, persist. Science doesn’t remove mysteries from our lives, it unearths them. And the ones it finds speak to us, saying: There is more to everything than meets the eye.