Gödel, Escher, Bach: An Eternal Golden Braid (69 page)

without thereby becoming a different machine, namely the old machine with a "new part"

added. Btu it is inherent in our idea of a conscious mind that it can reflect upon itself and criticize its own performances, and no extra part is required to (to this: it is already complete, and has no Achilles' heel.

The thesis thus begins to become more of a matter of conceptual analysis than mathematical discovery. This is borne out by considering another argument put forward by Turing. So far, we have constructed only fairly simple and predictable artifacts. When we increase the complexity of our machines, there may, perhaps, be surprises in store for us.

He draws a parallel with a fission pile. Below a certain "critical" size, nothing much happens: but above the critical size, the sparks begin to fly. So too, perhaps, with brains and machines. Most brains and all machines are, at present, sub-critical"-they react to incoming stimuli in a stodgy and uninteresting way, have no ideas of their own, can produce only stock responses-but a few brains at present, and possibly some machines in the future, are super-critical, and scintillate on their own account. Turing is suggesting that it is only a matter of complexity, and that above a certain level of complexity a qualitative difference appears, so that "super-critical" machines will be quite unlike the simple ones hitherto envisaged.

This may be so. Complexity often does introduce qualitative differences. Although it sounds implausible, it might turn out that above a certain level of complexity, a machine ceased to be predictable, even in principle, and started doing things on its own account, or, to use a very revealing phrase, it might begin to have a mind of its own. It might begin to have a mind of its own. It would begin to have a mind of its own when it was no longer entirely predictable and entirely docile, but was capable of doing things which we recognized as intelligent, and not just mistakes or random shots, but which we had not programmed into it. But then it would cease to be a machine, within the meaning of the act.

What is at stake in the mechanist debate is not how minds are, or might be, brought into being, but how they operate. It is essential for the mechanist thesis that the mechanical model of the mind shall operate according to "mechanical principles," that is, that we can understand the operation of the whole in terms of the operations of its parts, and the operation of each part either shall be determined by its initial state and the construction of the machine, or shall be a random choice between a determinate number of determinate operations. If the mechanist produces a machine which is so complicated that this ceases to hold good of it, then it is no longer a

machine for the purposes of our discussion, no matter how it was constructed. We should say, rather, that he had created a mind, in the same sort of sense as we procreate people at present. There would then be two ways of bringing new minds into the world, the traditional way, by begetting children born of women, and a new way by constructing very, very complicated systems of, say, valves and relays. When talking of the second way. we should take care to stress that although what was created looked like a machine, it was not one really, because it was not just the total of its parts. One could not tell what it was going to do merely by knowing the way in which it was built up and the initial state of its parts: one could not even tell the limits of what it could do, for even when presented with a Gödel-type question, it got the answer right. In fact we should say briefly that any system which was not floored by the Gödel question was eo ipso not a Turing machine, i.e. not a machine within the meaning of the act .3

In reading this passage, my mind constantly boggles at the rapid succession of topics, allusions, connotations, confusions, and conclusions. We jump from a Carrollian paradox to Gödel to Turing to Artificial Intelligence to holism and reductionism, all in the span of two brief pages. About Lucas one can say that he is nothing if not stimulating. In the following Chapters, we shall come back to many of the topics touched on so tantalizingly and fleetingly in this odd passage.

Aria with Diverse Variations

Achilles has been unable to 'sleep these past few nights. His friend the
Tortoise has come over tonight, to keep him company during these annoying
hours.

Tortoise: I am so sorry to hear of the troubles that have been plaguing you, my dear Achilles. I hope my company will provide a welcome relief from all the unbearable stimulation which has kept you awake. Perhaps I will bore you sufficiently that you will at long last go to sleep. In that way, I will be of some service.

Achilles: Oh, no, I am afraid that I have already had some of the world's finest bores try their hand at boring me to sleep-and all, sad to say, to no avail. So you will be no match for them. No, Mr. T, I invited you over hoping that perhaps you could entertain me with a little this or that, taken from number theory, so that I could at least while away these long hours in an agreeable fashion. You see, I have found that a little number theory does wonders for my troubled psyche.

Tortoise: How quaint an idea! You know, it reminds me, just a wee bit, of the story of poor Count Kaiserling.

'Achilles: Who was he?

Tortoise: Oh, he was a Count in Saxony in the eighteenth century-a Count of no account, to tell the truth-but because of him-well, shall I tell you the story? It is quite entertaining.

Achilles: In that case, by all means, do!

Tortoise: There was a time when the good Count was suffering from sleeplessness, and it just so happened that a competent musician lived in the same town, and so Count Kaiserling commissioned this musician to compose a set of variations to be played by the Count's court harpsichordist for him during his sleepless nights, to make the hours pass by more pleasantly.

Achilles: Was the local composer up to the challenge?

Tortoise: I suppose so, for after they were done, the Count rewarded him most lucratively-he presented him with a gold goblet containing one hundred Louis d'or.

Achilles: You don't say! I wonder where he came upon such a goblet and all those Louis d'or, in the first place.

Tortoise. Perhaps he saw it in a museum, and took a fancy to it.

Achilles: Are you suggesting he absconded with it?

Tortoise: Now, now, I wouldn't put it exactly that way, but ... Those days, Counts could get away with most anything. Anyway, it is clear that the Count was most pleased with the music, for he was constantly entreating his harpsichordist-a mere lad of a fellow, name of Goldberg-to

play one or another of these thirty variations. Consequently (and somewhat ironically) the variations became attached to the name of young Goldberg, rather than to the distinguished Count's name.

Achilles: You mean, the composer was Bach, and these were the so-called "Goldberg Variations"?

Tortoise: Do I ever! Actually, the work was entitled Aria with Diverse Variations, of which there are thirty. Do you know how Bach structured these thirty magnificent variations?

Achilles: Do tell.

Tortoise: All the pieces-except the final one-are based on a single theme, which he called an "aria". Actually, what binds them all together is not a common melody, but a common harmonic ground. The melodies may vary, but underneath, there is a constant theme. Only in the last variation did Bach take liberties. It is a sort of "post-ending ending". It contains extraneous musical ideas having little to do with the original Theme-in fact, two German folk tunes. That variation is called a

"quodlibet".

Achilles: What else is unusual about the Goldberg Variations?

Tortoise: Well, every third variation is a canon. First a canon in two canonizing voices enter on the SAME note. Second, a canon in which one of the canonizing voices enters ONE NOTE HIGHER than the first. Third, one voice enters Two notes higher than the other. And so on, until the final canon has entries just exactly one ninth apart. Ten canons, all told. And

Achilles: Wait a minute. Don't I recall reading somewhere or other about fourteen recently discovered Goldberg canons ...

Tortoise: Didn't that appear in the same journal where they recently reported the discovery of fourteen previously unknown days in November?

Achilles: No, it's true. A fellow named Wolff-a musicologist-heard about a special copy of the Goldberg Variations in Strasbourg. He went there to examine it, and to his surprise, on the back page, as a sort of "post-ending ending", he found these fourteen new canons, all based on the first eight notes of the theme of the Goldberg Variations. So now it is known that there are in reality forty-four Goldberg Variations, not thirty.

Tortoise: That is, there are forty-four of them, unless some other musicologist discovers yet another batch of them in some unlikely spot. And although it seems improbable, it is still possible, even if unlikely, that still another batch will be discovered, and then another one, and on and on and on ... Why, it might never stop! We may never know if or when we have the full complement of Goldberg Variations.

Achilles: That is a peculiar idea. Presumably, everybody thinks that this latest discovery was just a fluke, and that we now really do have all the Goldberg Variations. But just supposing that you are right, and some turn up sometime, we shall start to expect this kind of thing. At

that point, the name "Goldberg Variations" will start to shift slightly in meaning, to include not only the known ones, but also any others which might eventually turn up.

Their number-call it 'g'-is certain to be finite, wouldn't you agree?-but merely knowing that g is finite isn't the same as knowing how big g is. Consequently, this information won't tell us when the last Goldberg Variation has been located.

Tortoise: That is certainly true.

Achilles: Tell me-when was it that Bach wrote these celebrated variations?

Tortoise: It all happened in the year 1742, when he was Cantor in Leipzig. Achilles: 1742? Hmm ... That number rings a bell.

Tortoise: It ought to, for it happens to be a rather interesting number, being a sum of two odd primes: 1729 and 13.

Achilles: By thunder! What a curious fact' I wonder how often one runs across an even number with that property. Let's see

6= 3+3

8= 3+5

10= 3+7=

5+5

12= 5+7

14=3+11=

7+7

16=3+ 13= 5+ 11

18=5+ 13= 7+ 11

20=3+ 17= 7+ 13

22 = 3 + 19 = 5 + 17 =

11

+ 11

24=5+19=

7+17=11+13

26=3+23=

7+19=13+13

28 = 5 + 23 = 11 + 17

30 = 7 + 23 = 11 + 19 = 13 + 17

Now what do you know-according to my little table here, it seems to be quite a common occurrence. Yet I don't discern any simple regularity in the table so far.

Tortoise: Perhaps there is no regularity to be discerned.

Achilles: But of course there is! I am just not clever enough to spot it right off the bat.

Tortoise: You seem quite convinced of it.

Achilles: There's no doubt in my mind. I wonder ... Could it be that ALL even numbers (except 4) can be written as a sum of two odd primes?

Tortoise: Hmm ... That question rings a bell ... Ah, I know why! You're not the first person to ask that question. Why, as a matter of fact, in the year 1742, a mathematical amateur put forth this very question in a

Achilles: Did you say 1742? Excuse me for interrupting, but I just noticed that 1742

happens to be a rather interesting number, being a difference of two odd primes: 1747 and 5.

Tortoise: By thunder! What a curious fact! I wonder how often one runs across an even number with that property.

Achilles: But please don't let me distract you from your story.

Tortoise: Oh, yes-as I was saying, in 1742. a certain mathematical amateur, whose name escapes me momentarily, sent a letter to Euler, who at the time was at the court of King Frederick the Great in Potsdam, and-well, shall I tell you the story? It is not without charm. Achilles: In that case, by all means, do!

Tortoise: Very well. In his letter, this dabbler in number theory propounded an unproved conjecture to the great Euler: "Every even number can he represented as a sum of two odd primes." Now what was that fellow's name?

Achilles: I vaguely recollect the story, from some number theory book or other. Wasn't the fellow named Iiupfergiidel

Tortoise: Hmm ... No, that sounds too long.

Achilles: Could it have been "Silberescher"?

Tortoise: No, that's not it, either. There's a name on the tip of' my tongue-ah-ah-oh yes! It was "Goldbach"! Goldbach was the fellow. Achilles: I knew it was something like that.

Tortoise: Yes-your guesses helped jog my memory. It's quite odd, how one occasionally has to hunt around in one's memory as if for a book in a library without call numbers

... But let us get back to 1742.

Achilles: Indeed, let's. I wanted to ask you: did Euler ever prove that this guess by Goldbach was right?

Tortoise: Curiously enough, he never even considered it worthwhile working on.

However, his disdain was not shared by all mathematicians. In fact, it caught the fancy of many, and became known as the "Goldbach Conjecture".

Achilles: Has it ever been proven correct?

Tortoise: No, it hasn't. But there have been some remarkable near misses. For instance, in 1931 the Russian number theorist Schnirelmann proved that any number-even or odd-can be represented as the sum of not more than 300,000 primes.