Gödel, Escher, Bach: An Eternal Golden Braid (35 page)

How Universal Is DNA's Message?

The general question which we are facing, and which is very similar t questions inspired by the two plaques, is this: "How much of the co necessary for its own understanding is a message capable of restoring? can now revert to the original biological meanings of

"genotype" "phenotype"-
DNA
and a living organism-and ask similar quest Does DNA have universal triggering power? Or does it need a "biojukebox" to reveal its meaning?

Can
DNA
evoke a phenotype without being embedded in the proper chemical context?

To this question to answer is no-but a qualified no. Certainly a molecule of DNA in a vacuum will not create anything at all. However, if a molecule of
DNA
were set to seek its fortune in the universe, as we imagined the
BACH
and the
CAGE
were, it might be intercepted by an intelligent civilization. They might first of all recognize its frame message. Given that, they might to try to deduce from its chemical structure what kind of chemical environment it seemed to want, and then supply such an environment. Succes-

sively more refined attempts along these lines might eventually lead to a full restoration of the chemical context necessary for the revelation of
DNA's
phenotypical meaning.

This may sound a little implausible, but if one allows many millions of years for the experiment, perhaps the
DNA
's meaning would finally emerge.

On the other hand, if the sequence of bases which compose a strand of
DNA
were sent as abstract symbols (as in Fig. 41), not as a long helical molecule, the odds are virtually nil that this, as an outer message, would trigger the proper decoding mechanism which would enable the phenotype to be drawn out of the genotype. This would be a case of wrapping an inner message in such an abstract outer message that the context-restoring power of the outer message would be lost, and so in a very pragmatic sense, the set of symbols would have no intrinsic meaning. Lest you think this all sounds hopelessly abstract and philosophical, consider that the exact moment when phenotype can be said to be "available", or "implied", by genotype, is a highly charged issue in our day: it is the issue of abortion.

FIGURE 41. This Giant Aperiodic Crystal is the base sequence for the chromosome of
bacteriophage OX174. It is the first complete genome ever mapped out for any organism.

About 2,000 of these boustrophedonic pages would be needed to show the base sequence
of a single E. Coli cell, and about one million pages to show the base sequence of the
DNA of a single human cell. The book now in your hands contains roughly the same
amount of information as a molecular blueprint for one measly E. Coli cell.

Chromatic Fantasy, And Feud.

Having had a splendid dip in the pond, the Tortoise is just crawling out and
shaking himself dry, when who but Achilles walks by.

Tortoise: Ho there, Achilles. I was just thinking of you as I splash around in the pond.

Achilles: Isn't that curious? I was just thinking of you, too, while I meandered through the meadows. They're so green at this time of year.

Tortoise: You think so? It reminds me of a thought I was hoping to share with you.

Would you like to hear it?

Achilles: Oh, I would be delighted. That is, I would be delighted as long you're not going to try to snare me in one of your wicked traps of log Mr. T.

Tortoise: Wicked traps? Oh, you do me wrong. Would I do anything wicked? I'm a peaceful soul, bothering nobody and leading a gent; herbivorous life. And my thoughts merely drift among the oddities and quirks of how things are (as I see them). I, humble observer phenomena, plod along and puff my silly words into the air rather unspectacularly, I am afraid. But to reassure you about my intention I was only planning to speak of my Tortoise-shell today, and as you know, those things have nothing-nothing whatsoever-to do with logic!

Achilles: Your words Do reassure me, Mr. T. And, in fact, my curiosity quite piqued. I would certainly like to listen to what you have to say even if it is unspectacular.

Tortoise: Let's see ... how shall I begin? Hmm ... What strikes you me about my shell, Achilles?

Achilles: It looks wonderfully clean!

Tortoise: Thank you. I just went swimming and washed off several layers of dirt which had accumulated last century. Now you can see ho green my shell is.

Achilles: Such a good healthy green shell, it's nice to see it shining in sun.

Tortoise: Green? It's not green.

Achilles: Well, didn't you just tell me Tortoise: I did.

Achilles: Then, we agree: it is green. Tortoise: No, it isn't green.

Achilles: Oh, I understand your game. You're hinting to me that what you say isn't necessarily true; that Tortoises play with language; that your statements and reality don't necessarily match; that --

Tortoise: I certainly am not. Tortoises treat words as sacred. Tortoises revere accuracy.

Achilles: Well, then, why did you say that your shell is green, and that it is not green also?

Tortoise: I never said such a thing; but I wish I had. Achilles: You would have liked to say that?

Tortoise: Not a bit. I regret saying it, and disagree wholeheartedly with it. Achilles: That certainly contradicts what you said before!

Tortoise: Contradicts? Contradicts? I never contradict myself. It's not part of Tortoise-nature.

Achilles: Well, I've caught you this time, you slippery fellow, you. Caught you in a full-fledged contradiction.

Tortoise: Yes, I guess you did.

Achilles: There you go again! Now you're contradicting yourself more and more! You are so steeped in contradiction it's impossible to argue with you!

Tortoise: Not really. I argue with myself without any trouble at all. Perhaps the problem is with you. I would venture a guess that maybe you're the one who's contradictory, but you're so trapped in your own tangled web that you can't see how inconsistent you're being.

Achilles: What an insulting suggestion! I'm going to show you that you're the contradictory one, and there are no two ways about it.

Tortoise: Well, if it's so, your task ought to be cut out for you. What could be easier than to point out a contradiction? Go ahead-try it out.

Achilles: Hmm ... Now I hardly know where to begin. Oh ... I know. You first said that (1) your shell is green, and then you went on to say that (2) your shell is not green. What more can I say?

Tortoise: Just kindly point out the contradiction. Quit beating around the bush.

Achilles: But-but-but ... Oh, now I begin to see. (Sometimes I am so slow-witted!) It must be that you and I differ as to what constitutes a contradiction. That's the trouble.

Well, let me make myself very clear: a contradiction occurs when somebody says one thing and denies it at the same time.

Tortoise: A neat trick. I'd like to see it done. Probably ventriloquists would excel at contradictions, speaking out of both sides of their mouth, as it were. But I'm not a ventriloquist.

Achilles: Well, what I actually meant is just that somebody can say one thing and deny it all within one single sentence! It doesn't literally have to be in the same instant.

Tortoise: Well, you didn't give ONE sentence. You gave TWO.

Achilles: Yes-two sentences that contradict each other!

Tortoise: I am sad to see the tangled structure of your thoughts becoming so exposed, Achilles. First you told me that a contradiction is some thing which occurs in a single sentence. Then you told me that you

Found a contradiction in a pair of sentences I uttered. Frankly, it’s just as I said. Your own system of thought is so delusional that you manage to avoid seeing how inconsistent it is. From the outside, however plain as day.

Achilles: Sometimes I get so confused by your diversionary tactics tl can't quite tell if we're arguing about something utterly petty, or something deep and profound!

Tortoise: I assure you, Tortoises don't spend their time on the petty. Hence it's the latter.

Achilles: I am very reassured. Thank you. Now I have had a moment to reflect, and I see the necessary logical step to convince you that you contradicted yourself.

Tortoise: Good, good. I hope it's an easy step, an indisputable one.

Achilles: It certainly is. Even you will agree with it. The idea is that you believed sentence 1 ("My shell is green"), AND you believed sentence 2 ("My shell is not green"), you would believe one compound( sentence in which both were combined, wouldn't you?

Tortoise: Of course. It would only be reasonable ... providing just that the manner of combination is universally acceptable. But I'm sure we'll agree on that.

Achilles: Yes, and then I'll have you! The combination I propose is Tortoise: But we must be careful in combining sentences. For instance you'd grant that

"Politicians lie" is true, wouldn't you?

Achilles: Who could deny it?

Tortoise: Good. Likewise, "Cast-iron sinks" is a valid utterance, isn't it?

Achilles: Indubitably.

Tortoise: Then, putting them together, we get "Politicians lie in cast iron sinks". Now that's not the case, is it?

Achilles: Now wait a minute ... "Politicians lie in cast-iron sinks?" N no, but Tortoise: So, you see, combining two true sentences in one is not a policy, is it?

Achilles: But you-you combined the two-in such a silly way!

Tortoise: Silly? What have you got to object to in the way I combined them Would you have me do otherwise?

Achilles: You should have used the word "and", not "in".

Tortoise: I should have? You mean, if YOU'D had YOUR way, I should h; Achilles: No-it's the LOGICAL thing to do. It's got nothing to do with personally.

Tortoise: This is where you always lose me, when you resort to your L and its high-sounding Principles. None of that for me today, plea

Achilles: Oh, Mr. Tortoise, don't put me through all this agony. You k very well that that's what "and" means! It's harmless to combine true sentences with "and"!

Tortoise: "Harmless", my eye! What gall! This is certainly a pernicious plot to entrap a poor, innocent, bumbling Tortoise in a fatal contradiction. If it were so harmless, why would you be trying so bloody hard to get me to do it? Eh?

Achilles: You've left me speechless. You make me feel like a villain, where I really had only the most innocent of motivations.

Tortoise: That's what everyone believes of himself...

Achilles: Shame on me-trying to outwit you, to use words to snare you in a self-contradiction. I feel so rotten.

Tortoise: And well you should. I know what you were trying to set up. Your plan was to make me accept sentence 3, to wit: "My shell is green and my shell is not green".

And such a blatant falsehood is repellent to the Tongue of a Tortoise.

Achilles: Oh, I'm so sorry I started all this.

Tortoise: You needn't be sorry. My feelings aren't hurt. After all, I'm used to the unreasonable ways of the folk about me. I enjoy your company, Achilles, even if your thinking lacks clarity.

Achilles: Yes ... Well, I fear I am set in my ways, and will probably continue to err and err again, in my quest for Truth.

Tortoise: Today's exchange may have served a little to right your course. Good day, Achilles.

Achilles: Good day, Mr. T.

Words and Symbols

THE PRECEDING DIALOGUE is reminiscent of the
Two-Part Invention
by Lewis Carroll. In both, the Tortoise refuses to use normal, ordinary in the normal, ordinary way-or at least he refuses to do so when it is his advantage to do so. A way to think about the Carroll paradox was given last Chapter. In this Chapter we are going to make symbols dc Achilles couldn't make the Tortoise do with his words. That is, we are to make a formal system one of whose symbols will do just what A wished the word ànd' would do, when spoken by the Tortoise, and ail of whose symbols will behave the way the words
'if... then

.
. .' ought to behave. There are only two other words which we will attempt to deal with òr' and `not'. Reasoning which depends only on correct usage of these words is termed
propositional reasoning
.

Alphabet and First Rule of the Propositional Calculus

I will present this new formal system, called the Propositional Calculus, like a puzzle, not explaining everything at once, but letting you things out to some extent. We begin with the list of symbols:

< >
P Q R ´

∧
∨
⊃
~

The first rule of this system that I will reveal is the following: RULE OF JOINING: If x and y are theorems of the system, then so is the string < x∧y >.

This rule takes two theorems and combines them into one. It s remind you of the Dialogue.