Douglas Hofstadter Quotes

It turns out that an eerie type of chaos can lurk just behind a facade of order - and yet, deep inside the chaos lurks an even eerier type of order.

Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law

Meaning lies as much in the mind of the reader as in the Haiku.

Sometimes it seems as though each new step towards AI, rather than producing something which everyone agrees is real intelligence, merely reveals what real intelligence is not.

In the end, we self-perceiving, self-inventing, locked-in mirages are little miracles of self-reference.

A mirror mirroring a mirror

The paraphrase of Gödel's Theorem says that for any record player, there are records which it cannot play because they will cause its indirect self-destruction.

How gullible are you? Is your gullibility located in some "gullibility center" in your brain? Could a neurosurgeon reach in and perform some delicate operation to lower your gullibility, otherwise leaving you alone? If you believe this, you are pretty gullible, and should perhaps consider such an operation.

This idea that there is generality in the specific is of far-reaching importance.

I would like to understand things better, but I don’t want to understand them perfectly.

[...] provability is a weaker notion than truth

I wish my wish would not be granted!

For now, what is important is not finding the answer, but looking for it.

Some of us, perhaps all of us, believe that it is legitimate to kill enemy soldiers in a war, as if war were a special circumstance that shrinks the sizes of enemy souls.

We don't want to focus on the trees (or their leaves) at the expense of the forest.

Saying that studying the brain is limited to the study of physical entities would be like saying that literary criticism must focus on paper and bookbinding, ink and its chemistry, page sizes and margin widths, typefaces and paragraph lengths, and so forth.

We are all egocentric, and what is realest to each of us, in the end, is ourself.

No, no - I think about thinking

Deep understanding of causality sometimes requires the understanding of very large patterns and their abstract relationships and interactions, not just the understanding of microscopic objects interacting in microscopic time intervals.

What is an "I", and why are such things found (at least so far) only in association with, as poet Russell Edson once wonderfully phrased it, "teetering bulbs of dread and dream" -- that is, only in association with certain kinds of gooey lumps encased in hard protective shells mounted atop mobile pedestals that roam the world on pairs of slightly fuzzy, jointed stilts?

I enjoy acronyms. Recursive Acronyms Crablike "RACRECIR" Especially Create Infinite Regress

The key question is, no matter how much you absorb of another person, can you have absorbed so much of them that when that primary brain perishes, you can feel that that person did not totally perish from the earth... because they live on in a 'second neural home'?... In the wake of a human being's death, what survives is a set of afterglows, some brighter and some dimmer, in the collective brains of those who were dearest to them... Though the primary brain has been eclipsed, there is, in those who remain... a collective corona that still glows.

It is curious, how one often mistrusts one’s own opinions if they are stated by someone else.

Since, as is well know, God helps those who help themselves, presumably the Devil helps all those, and only those, who don't help themselves. Does the Devil help himself?

Please, Oh please, publish me in your collection of self-referential sentences!

It now becomes clear that consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it. By the same token, inconsistency is not an intrinsic property of any formal system.

Supperational thinkers, by recursive definition, include in their calculations the fact that they are in a group of superrational thinkers.

A computer program can modify itself but it cannot violate its own instructions — it can at best change some parts of itself by *obeying* its own instructions.

In the wake of a human being's death, what survives is a set of afterglows, some brighter and some dimmer, in the collective brains of those who were dearest to them...Though the primary brain has been eclipsed, there is, in those who remain...a collective corona that still glows. - Douglas Hofstadter

Now what is "music"–a sequence of vibrations in the air, or a succession of emotional responses in the brain?

But finally I realized that to me, Godel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to construct the central object, and came up with this book.

I personally cannot imagine that consciousness will be fully understood without reference to Godelian loops or level-crossing feedback loops.

Why is some music so much deeper and more beautiful than other music? It is because form, in music, is expressive–expressive to some strange subconscious regions of our minds. The sounds of music do not refer to serfs or city-states, but they do trigger clouds of emotion in our innermost selves; in that sense musical meaning IS dependent on intangible links from symbols to things in the world–those 'things', in this case, being secret software structures in our minds.

Perhaps the greatest contradiction in our lives, the hardest to handle, is the knowledge 'There was a time when I was not alive, and there will come a time when I am not alive.' On one level, when you 'step out of yourself' and see yourself as 'just another human being', it makes complete sense. But on another level, perhaps a deeper level, personal nonexistence makes no sense at all. All that we know is embedded inside our minds, and for all that to be absent from the universe is not comprehensible. This is a basic undeniable problem of life...

Irrationality is the square root of all evil.

To paraphrase Descartes again: "I think; therefore I have no access to the level where I sum.

We human beings are macroscopic structures in a universe whose laws reside at a microscopic level. As survival-seeking beings, we are driven to seek efficient explanations that make reference only to entities at our own level. We therefore draw conceptual boundaries around entities that we easily perceive, and in so doing we carve out what seems to us to be reality.

Concepts in the brains of humans acquired the property that they could get rolled together with other concepts into larger packets, and any such larger packet could then become a new concept in its own right. In other words, concepts could nest inside each other hierarchically, and such nesting could go on to arbitrary degrees.

And one of my firmest conclusions is that we always think by seeking and drawing parallels to things we know from our past, and that we therefore communicate best when we exploit examples, analogies, and metaphors galore, when we avoid abstract generalities, when we use very down-to-earth, concrete, and simple language, and when we talk directly about our own experience.

The theorem can be likened to a pearl, and the method of proof to an oyster. The pearl is prized for its luster and simplicity; the oyster is a complex living beast whose innards give rise to this mysteriously simple gem.

It sometimes feels as if I had shouted a deeply cherished message out into an empty chasm and nobody heard me.

My feeling is that the concept of superrationality is one whose truth will come to dominate among intelligent beings in the universe simply because its adherents will survive certain kinds of situations where its opponents will perish. Let’s wait a few spins of the galaxy and see. After all, healthy logic is whatever remains after evolution’s merciless pruning.

This computer-generated pangram contains six a's, one b, three c's, three d's, thirty-seven e's, six f's, three g's, nine h's, twelve i's, one j, one k, two l's, three m's, twenty-two n's, thirteen o's, three p's, one q, fourteen r's, twenty-nine s's, twenty-four t's, five u's, six v's, seven w's, four x's, five y's, and one z.

People enjoy inventing slogans which violate basic arithmetic but which illustrate “deeper” truths, such as “1 and 1 make 1” (for lovers), or “1 plus 1 plus 1 equals 1” (the Trinity). You can easily pick holes in those slogans, showing why, for instance, using the plus-sign is inappropriate in both cases. But such cases proliferate. Two raindrops running down a window-pane merge; does one plus one make one? A cloud breaks up into two clouds -more evidence of the same? It is not at all easy to draw a sharp line between cases where what is happening could be called “addition”, and where some other word is wanted. If you think about the question, you will probably come up with some criterion involving separation of the objects in space, and making sure each one is clearly distinguishable from all the others. But then how could one count ideas? Or the number of gases comprising the atmosphere? Somewhere, if you try to look it up, you can probably fin a statement such as, “There are 17 languages in India, and 462 dialects.” There is something strange about the precise statements like that, when the concepts “language” and “dialect” are themselves fuzzy.

A term meant to convey a person's inability to make sense of the numbers that run their lives.

There exist formal systems whose negative space (set of non-theorems) is not the positive space (set of theorems) of any formal system.

There are those who will immediately be drawn to the idea of pattern-seeking, and

No one who's lived and known reflection Could help but scorn the human host. No one who's sampled life's complexion Could fail to fear his dead past's ghost.

One must distinguish, it seems to me, between “head pattern” and “heart pattern”, or, in more objective-sounding terms, between syntactic pattern and semantic pattern.

It is an inherent property of intelligence that it can jump out of the task which it is performing, and survey what it has done; it is always looking for, and often finding, patterns.

Poised midway between the unvisualizable cosmic vastness of curved spacetime and the dubious shadowy flickerings of charged quanta, we human beings, more like rainbows and mirages than like raindrops or boulders, are unpredictable self-writing poems - vague, metaphorical, ambiguous, sometimes exceedingly beautiful.

The perception of an isomorphism between two known structures is a significant advance in knowledge-and I claim that it is such perceptions of isomorphism which create meanings in the minds of people.

We should have great respect for what seem like the most mundane of analogies, for when they are examined, they often can be seen to have sprung from, and to reveal, the deepest roots of human cognition.

I can't help but recall, at this point, a horribly elitist but very droll remark by one of my favorite writers, the American "critic of the seven arts", James Huneker, in his scintillating biography of Frédéric Chopin, on the subject of Chopin's étude Op. 25, No. 11 in A minor, which for me, and for Huneker, is one of the most stirring and most sublime pieces of music ever written: “Small-souled men, no matter how agile their fingers, should avoid it.” "Small-souled men"?! Whew! Does that phrase ever run against the grain of American democracy! And yet, leaving aside its offensive, archaic sexism (a crime I, too, commit in GEB, to my great regret), I would suggest that it is only because we all tacitly do believe in something like Hueneker's' shocking distinction that most of us are willing to eat animals of one sort or another, to smash flies, swat mosquitos, fight bacteria with antibiotics, and so forth. We generally concur that "men" such as a cow, a turkey, a frog, and a fish all possess some spark of consciousness, some kind of primitive "soul" but by God, it's a good deal smaller than ours is — and that, no more and no less, is why we "men" feel that we have the perfect right to extinguish the dim lights in the heads of these fractionally-souled beasts and to gobble down their once warm and wiggling, now chilled and stilled protoplasm with limitless gusto, and not feel a trace of guilt while doing so.

what happens on the lower level is responsible for what happens on the higher level, it is nonetheless irrelevant to the higher level. The higher level can blithely ignore the processes on the lower level.

If words were nuts and bolts, people could make any bolt fit into any nut: they'd just squish the one into the other, as in some surrealistic painting where everything goes soft. Language, in human hands, becomes almost like a fluid, despite the coarse grain of its components.

And that is also the way the human mind works — by the compounding of old ideas into new structures that become new ideas that can themselves be used in compounds, and round and round endlessly, growing ever more remote from the basic earthbound imagery that is each language’s soil.

Consistent or inconsistent, no one is exempt from the mystery of the self. Probably we are all inconsistent. The world is just too complicated for a person to be able to afford the luxury of reconciling all of his beliefs with each other. Tension and confusion are important in a world where many decisions must be made quickly. Miguel de Unamuno once said, 'If a person never contradicts himself, it must be that he says nothing.' I would say that we all are in the same boast as the Zen master who, after contradicting himself several times in a row, said to the confused Doko, 'I cannot understand myself.'.

What does it matter if two brains are isomorphic, or quasi-isomorphic, or not isomorphic at all? The answer is that we have an intuitive sense that, although other people differ from us in important ways, they are still 'the same' as we are in some deep and important ways. It would be instructive to be able to pinpoint what this invariant core of human intelligence is, and then to be able to describe the kinds of 'embellishments' which can be added to it, making each one of us a unique embodiment of this abstract and mysterious quality called 'intelligence.

Kurt Gödel was the first person to realize and exploit the fact that the positive integers, though they might superficially seem to be very austere and isolated, in fact constitute a profoundly rich representational medium. They can mimic or mirror any kind of pattern.

We mortals are condemned not to speak at that level of no information loss. We necessarily simplify, and indeed, vastly so. But that sacrifice is also our glory. Drastic simplification is what allows us to reduce situations to their bare bones, to discover abstract essences, to put our fingers on what matters, to understand phenomena at amazingly high levels, to survive reliably in this world, and to formulate literature, art, music, and science.

In short, Godel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved.

Where there's a pattern, there's a reason.

For Kant, analogy was the wellspring of all creativity, and Nietzsche gave a famous definition of truth as “a mobile army of metaphors”.

in fact we all know many more categories than we know words.

No thought can be formed that isn’t informed by the past; or, more precisely, we think only thanks to analogies that link our present to our past.

Often compound words have drifted so far from their etymological roots that native speakers can easily miss what is right in front of their eyes.

Therefore Godel's theorem had an electrifying effect upon logicians, mathematicians, and philosophers interested in the foundations of mathematics, for it showed that no fixed system, no matter how complicated could represent the complexity of the whole numbers: 0,1,2,3,.....

in order to see that what they did was “exactly the same thing”, one has to ignore almost all the details of the two situations in order to extract from them one single shared essence.

Consider the fact that you are now experiencing death, in a curious way. Not being in Paris right now, you know what it is like to be dead in Paris. No lights, no sounds -- nothing. The same goes for Timbuctu. In fact, you are dead everywhere -- except for one small spot. Just think how close you are to being dead everywhere!

Seeing anything as waves suggests immediate knobs: wavelength, frequency, amplitude, speed, medium, and a host of other basic notions that define the essence of undularity. Seeing anything as particles suggests totally different knobs: mass, shape, radius, rotation, constituents, and a host of other basic notions that define the essence of corpuscularity.

Historically, people have been naive about what qualities, if mechanized, would undeniably constitute intelligence. Is intelligence an ability to integrate functions symbolically? If so, then AI already exists, since symbolic integration routines outdo the best people in most cases. If intelligence involves learning, creativity, emotional responses, a sense of beauty, a sense of self, then there is a long road ahead, and it may be that these will only be realized when we have totally duplicated a living brain.

(1) Blurting may be considered as the reciprocal substitution of semiotic material (dubbing) for a semiotic dialogical product in a dynamic reflexion. The human-written sentences are numbers (1) to 3; they were drawn from the contemporary journal Art-Language and are -- as far as I can tell-- completely serious efforts among literate and sane people to communicate something to each other.

Where There’s Pattern, There’s Reason The key thought in the preceding few lines is the article of faith that this pattern cannot merely be a coincidence. A mathematician who finds a pattern of this sort will instinctively ask, “Why? What is the reason behind this order?” Not only will all mathematicians wonder what the reason is, but even more importantly, they will all implicitly believe that whether or not anyone ever finds the reason, there must be a reason for it. Nothing happens “by accident” in the world of mathematics.

Just as we need to hide the massively complex details inside our fancy gadgets by elegant and user-friendly packaging, so we need to hide the details of many ideas in order to talk about them in a sufficiently compact way that we won’t get lost in a mountain of details.

Consistent or inconsistent, no one is exempt from the mystery of the self. Probably we are all inconsistent. Te world is just too complicated for a person to be able to afford the luxury of reconciling all of his beliefs with each other. Tension and confusion are important in a world where many decisions must be made quickly. Miguel de Unamuno once said, 'If a person never contradicts himself, it must be that he says nothing.' I would say that we all are in the same boast as the Zen master who, after contradicting himself several times in a row, said to the confused Doko, 'I cannot understand myself.

The repetition of the orphanage drama year after year, echoing the Nietzschean idea of eternal recurrence -- that everything that has happened will happen again and again -- seems to rob the little world of any real meaning. Why should the repetition of the fire inspector's lament make it sound so hollow?

Computers are ridiculous. So is science in general. CHURCH_TURING Thesis, Theodore Rosak Version This view is prevalent among certain people who see in anything smacking of numbers or exactitude a threat to human values. It is too bad that they do not appreciate the depth and complexity and beauty involved in exploring abstract structures sch as the human mind, where, indeed, one comes in intimate contact with the ultimate questions of what to be human is.

By the time we emerge from childhood, we have acquired a reflex-level intuition for where most of our everyday world’s loci of unpredictability lie, and the more unpredictable end of this spectrum simultaneously beckons to us and frightens us. We’re pulled by but fearful of risk-taking. That is the nature of life.

Through many types of abstraction and analogy-making and inductive reasoning, and through many long and tortuous chains of citations of all sorts of authorities (which constitute an indispensable pillar supporting every adult’s belief system, despite the insistence of high-school teachers who year after year teach that “arguments by authority” are spurious and are convinced that they ought to be believed because they are, after all, authority figures), we build up an intricate, interlocked set of beliefs as to what exists “out there” — and then, once again, that set of beliefs folds back, inevitably and seamlessly, to apply to our own selves.

We claim that cognition takes place thanks to a constant flow of categorizations, and that at the base of it all is found, in contrast to classification (which aims to put all things into fixed and rigid mental boxes), the phenomenon of categorization through analogy-making, which endows human thinking with its remarkable fluidity.

the Godelian strange loop that arises in formal systems in mathematics (i.e., collections of rules for churning out an endless series of mathematical truths solely by mechanical symbol-shunting without any regard to meanings or ideas hidden in the shapes being manipulated) is a loop that allows such a system to "perceive itself", to talk about itself, to become "self-aware", and in a sense it would not be going too far to say that by virtue of having such a loop, a formal system acquires a self.

We have a very hard time “seeing” our cognitive activity because it is the medium in which we swim. The attempt to put our finger on what counts in any given situation leads us at times to making connections between situations that are enormously different on their surface and at other times to distinguishing between situations that on first glance seem nearly identical. Our constant jockeying back and forth among our categories runs the gamut from the most routine behaviors to the most creative ones.

It was a hundred years later that Einstein gave a theory (general relativity) which said that the geometry of the universe is determined by its content of matter, so that no one geometry is intrinsic to space itself. Thus to the question, "Which geometry is true?" nature gives an ambiguous answer not only in mathematics, but also in physics

Douglas R. Hofstadter, an American researcher, speculates that the human mind has consciousness because it has the capability of self-reference. Since we can think about ourselves and think about ourselves thinking about ourselves, etc., we are capable of feeling that we are an "I". Contrast that with what we have learned in this chapter. This chapter tries to show that the computer's ability to perform self-reference is the cause of its limitations. Can we say that self-reference in computers brings limitations while in humans it causes consciousness? Perhaps.

When a system of "meaningless" symbols has patterns in it that accurately track, or mirror, various phenomena in the world, then that tracking or mirroring imbues the symbols with some degree of meaning-indeed, such tracking or mirroring is no less and no more than what meaning is. Depending on how complex and subtle and reliable the tracking is, different degrees of meaningfulness arise.

No one knows where the borderline between non-intelligent behavior and intelligent behavior lies; in fact, to suggest that a sharp borderline exists is probably silly. But essential abilities for intelligence are certainly: to respond to situations very flexibly; to take advantage of fortuitous circumstances; to make sense out of ambiguous or contradictory messages; to recognize the relative importance of different elements of a situation; to find similarities between situations despite differences which may separate them; to draw distinctions between situations despite similarities may link them; to synthesize new concepts by taking old them together in new ways; to come up with ideas which are novel.

It seems to me, therefore, that the instinctive although seldom articulated purpose of holding a funeral or memorial service is to reunite the people most intimate with the deceased, and to collectively rekindle in them all, for one last time, the special living flame that represents the essence of that beloved person, profiting directly or indirectly from the presence of one another, feeling the presence of that person in the brains that remain, and thus solidifying to the maximal extent possible those secondary personal gemmae that remain aflicker in all these different brains. Though the primary brain has been eclipsed, there is, in those who remain and who are gathered to remember and reactivate the spirit of the departed, a collective corona that still glows. This is what human love means. The word "love" cannot, thus, be separated from the word "I"; the more deeply rooted the symbol for someone inside you, the greater the love, the brighter the light that remains behind.

Such concepts, be they concrete or abstract, are selectively mobilized instant by instant, and nearly always without any awareness on our part, and it is this ceaseless activity that allows us to build up mental representations of situations we are in, to have complex feelings about them, and to have run-of-the-mill as well as more exalted thoughts. No thought can be formed that isn’t informed by the past; or, more precisely, we think only thanks to analogies that link our present to our past.

Thus our categories keep us well informed at all times, allowing us to bypass the need for direct observation. If we didn’t constantly extrapolate our knowledge into new situations — if we refrained from making inferences — then we would be conceptually blind. We would be unable to think or act, doomed to permanent uncertainty and to eternal groping in the dark. In short, in order to perceive the world around us, we depend just as much on categorization through analogy as we do on our eyes or our ears.

I am a lifelong lover of form–content interplay, and this book is no exception. As with several of my previous books, I have had the chance to typeset it down to the finest level of detail, and my quest for visual elegance on each page has had countless repercussions on how I phrase my ideas. To some this may sound like the tail wagging the dog, but I think that attention to form improves anyone’s writing. I hope that reading this book not only is stimulating intellectually but also is a pleasant visual experience.

It is numbers like 545 billion that we are dealing with when we talk about a Defense Department overrun of $750 billion for the next four years. A really fancy single-user computer (the kind I wouldn’t mind having) costs approximately $75,000. With $750 billion to throw around, we could give one to every person in New York City, which is to say, we could buy about ten million of them. Or, we could give $1 million to every person in San Francisco, and still have enough left over to buy a bicycle for everyone in China! There’s no telling what good uses we could put $750 billion to. But instead, it will go into bullets and tanks and fighters and war games and missile systems and jet fuel and marching bands and so on. An interesting way to spend $750 billion, but I can think of better ways.

We believe in marbles that disintegrate when we search for them but that are as real as any genuine marble when we're not looking for them. Our very nature is such as to prevent us from fully understanding its very nature. Poised midway between the unvisualizable cosmic vastness of curved spacetime and the dubious, shadowy flickerings of charged quanta, we human beings, more like rainbows and mirages than like raindrops or boulders, are unpredictable self-writing poems—vague, metaphorical, ambiguous, and sometimes exceedingly beautiful. [...] What one gives up on is a childlike sense that things are exactly as they appear, and that our solid-seeming, marble-like "I" is the realest thing in the world; what one acquires is an appreciation of how tenuous we are at our cores, and how wildly different we are from what we seem to be.

Do you remember your first sip of beer? Terrible! How could anyone like that stuff? But beer, you reflect, is an acquired taste; one gradually trains oneself—or just comes—to enjoy that flavor. What flavor? The flavor of that first sip? No one could like that flavor! Beer tastes different to the experienced beer drinker. Then beer isn't an acquired tast; one doesn't learn to like that first taste; one gradually comes to experience a different, and likable, taste. Had the first sip tasted that way, you would have liked beer wholeheartedly from the beginning!

What gives us word-users the right to make life-and-death decisions concerning other living creatures that have no words? Why do we find ourselves in positions of such anguish (at least for some of us)? In the final analysis, it is simply because might makes right, and we humans, thanks to the intelligence afforded us by the complexity of our brains and our embeddedness in rich languages and cultures, are indeed high and mighty, relative to the “lower” animals (and vegetables). By virtue of our might, we are forced to establish some sort of ranking of creatures, whether we do so as a result of long and careful personal reflections or simply go along with the compelling flow of the masses. Are cows just as comfortably killable as mosquitoes? Would you feel any less troubled by swatting a fly preening on a wall than by beheading a chicken quivering on a block?

Is there some amazing rational thing you do when your mind's running in all different directions?" she managed. "My own approach is usually to identify the different desires, give them names, conceive of them as separate individuals, and let them argue it out inside my head. So far the main persistent ones are my Hufflepuff, Ravenclaw, Gryffindor, and Slytherin sides, my Inner Critic, and my simulated copies of you, Neville, Draco, Professor McGonagall, Professor Flitwick, Professor Quirrell, Dad, Mum, Richard Feynman, and Douglas Hofstadter." Hermione considered trying this before her Common Sense warned that it might be a dangerous sort of thing to pretend. "There's a copy of me inside your head?" "Of course there is!" Harry said. The boy suddenly looked a bit more vulnerable. "You mean there isn't a copy of me living in your head?" There was, she realized; and not only that, it talked in Harry's exact voice. "It's rather unnerving now that I think about it," said Hermione. "I do have a copy of you living in my head. It's talking to me right now using your voice, arguing how this is perfectly normal." "Good," Harry said seriously. "I mean, I don't see how people could be friends without that.

if one keeps climbing upward in the chain of command within the brain, one finds at the very top those over-all organizational forces and dynamic properties of the large patterns of cerebral excitation that are correlated with mental states or psychic activity…. Near the apex of this command system in the brain…. we find ideas. Man over the chimpanzee has ideas and ideals. In the brain model proposed here, the causal potency of an idea, or an ideal, becomes just as real as that of a molecule, a cell, or a nerve impulse. Ideas cause ideas and help evolve new ideas. They interact with each other and with other mental forces in the same brain, in neighboring brains, and, thanks to global communication, in far distant, foreign brains. And they also interact with the external surroundings to produce in toto a burst-wise advance in evolution that is far beyond anything to hit the evolutionary scene yet, including the emergence of the living cell. Who

Godel showed how a statement about any mathematical formal system (such as the assertion that Principia Mathematica is contradiction-free) can be translated into a mathematical statement inside number theory (the study of whole numbers). In other words, any metamathematical statement can be imported into mathematics, and in its new guise the statement simply asserts (as do all statements of number theory) that certain whole numbers have certain properties or relationships to each other. But on another level, it also has a vastly different meaning that, on its surface, seems as far removed from a statement of number theory as would be a sentence in a Dostoevsky novel.

It is an inherent property of intelligence that it can jump out of the task which it is performing, and survey what it has done; it is always looking for, and often finding, patterns. Now I said that an intelligence canjump out of its task, but that does not mean that it always will. However, a little prompt- ing will often suffice. For example, a human being who is reading a book may grow sleepy. Instead of continuing to read until the book is finished, he is just as likely to put the book aside and turn off the light. He has stepped "out of the system" and yet it seems the most natural thing in the world to us. Or, suppose person A is watching television when person B comes in the room, and shows evident displeasure with the situation. Person A may think he understands the problem, and try to remedy it by exiting the present system (that television program), and flipping the chan- nel knob, looking for a better show. Person B may have a more radical concept of what it is to "exit the system"-namely to turn the television off! Of course, there are cases where only a rare individual will have the vision to perceive a system which governs many peoples' lives, a system which had never before even been recognized as a system; then such people often devote their lives to convincing other people that the system really is there, and that it ought to be exited from!

An amusing way to see the incorrectness of Lucas' argument is to translate it into a battle between men and women ... In his wanderings, Loocus the Thinker one day comes across an unknown object—a woman. Such a thing he has never seen before, and at first he is wondrous thrilled at her likeness to himself; but then, slightly scared of her as well, he cries to all the men about him, "Behold! I can look upon her face, which is something she cannot do—therefore women can never be like me!" And thus he proves man's superiority over women, much to his relief, and that of his male companions. Incidentally, the same argument proves that Loocus is superior to all other males, as well—but he doesn't point that out to them. The woman argues back: "Yes, you can see my face, which is something I can't do—but I can see your face, which is something you can't do! We're even." However, Loocus comes up with an unexpected counter: "I'm sorry, you're deluded if you think you can seemy face. What you women do is not the same as what we men do—it is, as I have already pointed out, of an inferior caliber, and does not deserve to be called by the same name. You may call it 'womanseeing'. Now the fact that you can 'womansee' my face is of no import, because the situation is not symmetric. You see?" "I woman-see," womanreplies the woman, and womanwalks away...