Gödel, Escher, Bach: An Eternal Golden Braid (85 page)

Now with this wonderful Escherian metaphor, let us return to the program versus the human. We were talking about trying to encapsulate the "Gödelizing operator" inside the program itself. Well, even if we had written a program which carried the operation out, that program would not capture the essence of Gödel’s method. For once again, we, outside the system, could still "zap" it in a way which it couldn't do. But then are we arguing with, or against, Lucas

The Limits of Intelligent Systems

Against. For the very fact that we cannot write a program to do "Gödelizing" must make us somewhat suspicious that we ourselves could do it in every case. It is one thing to make the argument in the abstract that Gödelizing "can be done"; it is another thing to know how to do it in every particular case. In fact, as the formal systems (or programs) escalate in complexity, our own ability to "Gödelize" will eventually begin to waver. It must, since, as we have said above, we do not have any algorithmic way of describing how to perform it. If we can't tell explicitly what is involved in applying the Gödel method in all cases, then for each of us there will eventually come some case so complicated that we simply can't figure out how to apply it.

Of course, this borderline of one's abilities will be somewhat ill-defined, just as is the borderline of weights which one can pick up off the ground. While on some days you may not be able to pick up a 250-pound object, on other days maybe you can.

Nevertheless, there are no days whatsoever on which you can pick up a 250-ton object.

And in this sense, though everyone's Godelization threshold is vague, for each person, there are systems which lie far beyond his ability to Godelize.

This notion is illustrated in the Birthday Cantatatata. At first, it seems obvious that the Tortoise can proceed as far as he wishes in pestering Achilles. But then Achilles tries to sum up all the answers in a single swoop. This is a move of a different character than any that has gone before, and is given the new name 'co'. The newness of the name is quite important. It is the first example where the old naming scheme-which only included names for all the natural numbers-had to be transcended. Then come some more extensions, some of whose names seem quite obvious, others of which are rather tricky.

But eventually, we run out of names once again-at the point where the answer-schemas

, ωω, ωωω …..

are all subsumed into one outrageously complex answer schema. The altogether new name 'e„' is supplied for this one. And the reason a new name is needed is that some fundamentally new kind of step has been taken—a sort of irregularity has been encountered. Thus a new name must be applied ad hoc.

There Is No Recursive Rule for Naming Ordinals.

Now offhand you might think that these irregularities in the progression >m ordinal to ordinal (as these names of infinity are called) could be handled by a computer program.

That is, there would be a program to produce new names in a regular way, and when it ran out of gas, it would invoke the "irregularity handler", which would supply a new name, and pass control back to the simple one. But this will not work. It turns out that irregularities themselves happen in irregular ways, and one would need o a second-order program-that is, a program which makes new programs which make new names. And even this is not enough. Eventually, a third-order program becomes necessary. And so on, and so on.

All of this perhaps ridiculous-seeming complexity stems from a deep °theorem, due to Alonzo Church and Stephen C. Kleene, about the structure of these "infinite ordinals", which says:

There is no recursively related notation-system which gives a name to every constructive ordinal.

hat "recursively related notation-systems" are, and what "constructive ordinals" are, we must leave to the more technical sources, such as Hartley )gets' book, to explain. But the intuitive idea has been presented. As the ordinals get bigger and bigger, there are irregularities, and irregularities in e irregularities, and irregularities in the irregularities in the irregularities, etc. No single scheme, no matter how complex, can name all e ordinals.

And from this, it follows that no algorithmic method can tell w to apply the method of Gödel to all possible kinds of formal systems. ad unless one is rather mystically inclined, therefore one must conclude at any human being simply will reach the limits of his own ability to 5delize at some point. From there on out, formal systems of that complex, though admittedly incomplete for the Gödel reason, will have as much power as that human being.

Other Refutations of Lucas

Now this is only one way to argue against Lucas' position. There are others, possibly more powerful, which we shall present later. But this counterargument has special interest because it brings up the fascinating concept trying to create a computer program which can get outside of itself, see itself completely from the outside, and apply the Gödel zapping-trick to itself. Of course this is just as impossible as for a record player to be able to ay records which would cause it to break.

But-one should not consider
TNT
defective for that reason. If there a defect anywhere, it is not in
TNT
, but in our expectations of what it should he able to do. Furthermore, it is helpful to realize that we are equally vulnerable to the word trick which Gödel transplanted into mathematical formalisms: the Epimenides paradox. This was quite cleverly pointed out

by C. H. Whitely, when he proposed the sentence "Lucas cannot consistently assert this sentence." If you think about it, you will see that (1) it is true, and yet (2) Lucas cannot consistently assert it. So Lucas is also "incomplete" with respect to truths about the world. The way in which he mirrors the world in his brain structures prevents him from simultaneously being "consistent" and asserting that true sentence. But Lucas is no more vulnerable than any of us. He is just on a par with a sophisticated formal system.

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 see my 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 womansee," womanreplies the woman, and womanwalks away .. .

Well, this is the kind of "heads-in-the-sand" argument which you have to be willing to stomach if you are bent on seeing men and women running ahead of computers in these intellectual battles.

Self-Transcendence-A Modern Myth

It is still of great interest to ponder whether we humans ever can jump out of ourselves-or whether computer programs can jump out of themselves. Certainly it is possible for a program to modify itself-but such modifiability has to be inherent in the program to start with, so that cannot be counted as an example of "jumping out of the system". No matter how a program twists and turns to get out of itself, it is still following the rules inherent in itself. It is no more possible for it to escape than it is for a human being to decide voluntarily not to obey the laws of physics. Physics is an overriding system, from which there can be no escape. However, there is a lesser ambition which it is possible to achieve: that is, one can certainly Jump from a subsystem of one's brain into a wider subsystem. One can step out of ruts on occasion. This is still due to the interaction of various subsystems of one’s brain, but it can feel very much like stepping entirely out of oneself. Similarly, it is entirely conceivable that a partial ability to “step outside of itself” could be embodied in a computer program.

However, it is important to see the distinction between perceiving oneself, and transcending oneself. You can gain visions of yourself in all sorts of rays-in a mirror, in photos or movies, on tape, through the descriptions if others, by getting psychoanalyzed, and so on. But you cannot quite break out of your own skin and be on the outside of yourself (modern occult movements, pop psychology fads, etc. notwithstanding).
TNT

can talk about itself, but it cannot jump out of itself. 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. This is reminiscent of the numerous paradoxical question,

"Can God make a stone so heavy that he can’t lift it?"

Advertisement and Framing Devices

[his drive to jump out of the system is a pervasive one, and lies behind all progress in art, music, and other human endeavors. It also lies behind such trivial undertakings as the making of radio and television commercials. [his insidious trend has been beautifully perceived and described by Irving Goffman in his book
Frame Analysis
: For example, an obviously professional actor completes a commercial pitch and, with the camera still on him, turns in obvious relief from his task, now to take real pleasure in consuming the product he had been advertising.

This is, of course, but one example of the way in which TV and radio commercials are coming to exploit framing devices to give an appearance of naturalness that (it is hoped) will override the reserve auditors have developed.

Thus, use is currently being made of children's voices, presumably because these seem unschooled; street noises, and other effects to give the impression of interviews with unpaid respondents; false starts, filled pauses, byplays, and overlapping speech to simulate actual conversation; and, following Welles, the interception of a firm's jingle commercials to give news of its new product, alternating occasionally with interception by a public interest spot, this presumably keeping the faith of the auditor alive.

The more that auditors withdraw to minor expressive details as a test of genuineness, the more that advertisers chase after them. What results is a sort of interaction pollution, a disorder that is also spread by the public relations consultants of political figures, and, more modestly, by micro-sociology.'

Here we have yet another example of an escalating "
TC
-battle"-the antagonists this time being Truth and Commercials.

Simplicio, Salviati, Sagredo: Why Three?

There is a fascinating connection between the problem of jumping out of ie system and the quest for complete objectivity. When I read Jauch's four dialogues in Are Quanta Real? based on Galileo's four Dialogues Concerning Two New Sciences, I found myself wondering why there were three characters participating. Simplico, Salviati and Sagredo.

Why wouldn’t two have

sufficed: Simplicio, the educated simpleton, and Salviati, the knowledgeable thinker?

What function does Sagredo have? Well, he is supposed to be a sort of neutral third party, dispassionately weighing the two sides and coming out with a "fair" and "impartial"

judgment. It sounds very balanced, and yet there is a problem: Sagredo is always agreeing with Salviati, not with Simplicio. How come Objectivity Personified is playing favorites? One answer, of course, is that Salviati is enunciating correct views, so Sagredo has no choice. But what, then, of fairness or "equal time"?

By adding Sagredo, Galileo (and Jauch) stacked the deck
more
against Simplicio, rather than less. Perhaps there should be added a yet higher level Sagredo-someone who will be objective about this whole situation ... You can see where it is going. We are getting into a never-ending series of "escalations in objectivity", which have the curious property of never getting any more objective than at the first level: where Salviati is simply right, and Simplicio wrong. So the puzzle remains: why add Sagredo at all? And the answer is, it gives the illusion of stepping out of the system, in some intuitively appealing sense.

Zen and "Stepping Out"

In Zen, too, we can see this preoccupation with the concept of transcending the system.

For instance, the koan in which Tozan tells his monks that "the higher Buddhism is not Buddha". Perhaps, self-transcendence is even the central theme of Zen. A Zen person is always trying to understand more deeply what he is, by stepping more and more out of what he sees himself to be, by breaking every rule and convention which he perceives himself to be chained by-needless to say, including those of Zen itself. Somewhere along this elusive path may come enlightenment. In any case (as I see it), the hope is that by gradually deepening one's self-awareness, by gradually widening the scope of "the system", one will in the end come to a feeling of being at one with the entire universe.