Gödel, Escher, Bach: An Eternal Golden Braid (98 page)

0, and the mode is up. Then the program will copy itself next door in memory, with N = 1.

Repeating the process, the new program will self-rep next door to itself, with a copy having N = 2. And over and over again ... What happens is that a very large program is growing inside memory. When memory is full, the process quits. Now all of memory can be looked upon as being filled with one big program, composed of many similar, but differentiated, modules-or "cells". Now suppose we switch the mode to down, and run this big program.

What happens? The first "cell" runs, and calculates 1/1. The second "cell" runs, calculating -

1/3, and adding it to the previous result. The third "cell" runs, calculating + 1/5 and adding it on. .. The end result is that the whole "organism"-the big program-calculates the sum l -1/3 +1/5 -117 +1/9 -1/11 +1/13 -1/15 + .. .

to a large number of terms (as many terms as "cells" can fit inside memory). And since this series converges (albeit slowlv) to 7r/4, we have a "phenotype" whose function is to calculate the value of a famous mathematical constant.

Level Mixing in the Cell

I hope that the descriptions of processes such as labeling, self-assembly, differentiation, morphogenesis, as well as transcription and translation, have helped to convey some notion of the immensely complex system which is a cell-an information-processing system with some strikingly

novel features. We have seen, in the Central Dogmap, that although we can try to draw a clear line between program and data, the distinction is somewhat arbitrary. Carrying this line of thought further, we find that not only are program and data intricately woven together, but also the interpreter of programs, the physical processor, and even the language are included in this intimate fusion. Therefore, although it is possible (to some extent) to draw boundaries and separate out the levels, it is just as important-and ust as fascinating-to recognize the level-crossings and mixings. Illustrative of this is the amazing fact that in biological systems, all the various features necessary for self-rep (viz., language, program, data, interpreter, and processor) cooperate to such a degree that all of them are replicated simultaneously-which shows how much deeper is biological self-rep'ing than anything yet devised along those lines by humans. For instance, the self-rep program exhibited at the beginning of this Chapter takes for granted the pre-existence of three external aspects: a language, an interpreter, and a processor, and does not replicate those.

Let us try to summarize various ways in which the subunits of a cell can be classified in computer science terms. First, let us take DNA. Since DNA contains all the information for construction of proteins., which are the active agents of the cell, DNA can be viewed as a program written in a higher-level language, which is subsequently translated (or interpreted) into the "machine language" of the cell (proteins). On the other hand, DNA is itself a passive molecule which undergoes manipulation at the hands of various kinds of enzymes; in this sense, a DNA molecule is exactly like a long piece of data, as well. Thirdly, DNA contains the templates off of which the tRNA "flashcards" are rubbed, which means that DNA also contains the definition of its own higher-level language.

Let us move on to proteins. Proteins are active molecules, and carry out all the functions of the cell; therefore it is quite appropriate to think of them as programs in the

"machine language" of the cell (the cell itself being the processor). On the other hand, since proteins are hardware and most programs are software, perhaps it is better to think of the proteins as processors. Thirdly, proteins are often acted upon by other proteins, which means that proteins are often data. Finally, one can view proteins as interpreters; this involves viewing DNA as a collection of' high-level language programs, in which case enzymes are merely carrying out the programs written in the DNA code, which is to say, the proteins are acting as interpreters.

Then there are ribosomes and tRNA molecules. They mediate the translation from DNA to proteins, which can be compared to the translation of a program from a high-level language to a machine language; in other words, the ribosomes are functioning as interpreters and the tRNA molecules provide the definition of the higher-level language. But an alternative view of translation has it that the ribosomes are processors, while the tRNA's are interpreters.

We have barely scratched the surface in this analysis of interrelations between all these biomolecules. What we have seen is that nature feels quite

comfortable in mixing levels which we tend to see as quite distinct. Actually, in computer science there is already a visible tendency to nix all these seemingly distinct aspects of an information-processing system. This is particularly so in Artificial Intelligence research, which is usually at the forefront of computer language design.

The Origin of Life

A natural and fundamental question to ask, on learning of these incredibly intricately interlocking pieces of software and hardware is: "How did they ever get started in the first place?" It is truly a baffling thing. One has to imagine some sort of a bootstrap process occurring, somewhat like that which is used in the development of new computer languages-but a bootstrap from simple molecules to entire cells is almost beyond one's power to imagine. There are various theories on the origin of life. They all run aground on this most central of all central questions: "How did the Genetic Code, along with the mechanisms for its translation (ribosomes and tRNA molecules), originate" For the moment, we will have to content ourselves with a sense of wonder and awe, rather than with an answer. And perhaps experiencing that sense of wonder and awe is more satisfying than having an answer-at least for a while.

The Magnificrab, Indeed,

It is spring, and the Tortoise and Achilles are taking a Sunday promenade in
the woods together. They have decided to climb a hill at the top of which, it is
said, there is a wonderful teahouse, with all sorts of delicious pastries.

Achilles: Man oh man! If a crab

Tortoise: If a crab??

Achilles: I was about to say, if a crab ever were intelligent, then surely it would be our mutual friend the Crab. Why, he must be at least two times as smart as any crab alive.

Or maybe even three times as smart as any crab alive. Or perhaps

Tortoise: My soul! How you magnify the Crab!

Achilles: Well, I just happen to be an admirer of his ...

Tortoise: No need to apologize. I admire him, too. Speaking of Crab admirers, did I tell you about the curious fan letter which the Crab received not too long ago?

Achilles: I don't believe so. Who sent it?

Tortoise: It bore a postmark from India, and was from someone neither of us had ever heard of before-a Mr. Najunamar, I believe.

Achilles: I wonder why someone who never knew Mr. Crab would send him a letter-or for that matter, how they would get his address. Tortoise: Apparently whoever it was was under the illusion that the Crab is a mathematician. It contained numerous results, all of which were But, ho! Speak of the devil! Here comes Mr. Crab now, down the hill. Crab: Good-bye! It was nice to talk with you again. Well, I guess I had best be off. But I'm utterly stuffed-couldn't eat one more bite if I had to! I've just been up there myself-recommend it highly. Have you ever been to the teahouse at the crest of the hill? How are you, Achilles? Oh, there's Achilles. Hello, hello. Well, well, if it isn't Mr. T!

Tortoise: Hello, Mr. C. Are you headed up to the hilltop teahouse? Crab: Why, yes indeed, I am; how did you guess it? I'm quite looking forward to some of their special napoleons-scrumptious little morsels. I'm so hungry I could eat a frog. Oh, there's Achilles. How are you, Achilles?

Achilles: Could be worse, I suppose.

Crab: Wonderful! Well, don't let me interrupt your discussion. I'll just tag along.

Tortoise: Curiously enough, I was just about to describe your mysterious letter from that Indian fellow a few weeks back-but now that you're here. I'll let Achilles get the story from the Crab’s mouth.

FIGURE 104. Castrovalva, by'M. C. Escher (lithograph, 1930).

Crab: Well, it was this way. This fellow Najunamar had apparently never had any formal training in mathematics, but had instead worked out some of his own methods for deriving new truths of mathematics. Some of his discoveries defeated me completely; I had never seen anything in the least like them before. For instance, he exhibited a map of India that he had managed to color using no fewer than 1729 distinct colors.

Achilles: 1729! Did you say 1729? Crab: Yes-why do you ask?

Achilles: Well, 1729 is a very interesting number, you know. Crab: Indeed. I wasn't aware of it.

Achilles: In particular, it so happens that 1729 is the number of the taxicab which I took to Mr. Tortoise's this morning!

Crab: How fascinating! Could you possibly tell me the number of the trolley car which you'll take to Mr. Tortoise's tomorrow morning?

Achilles (after a moment's thought): It's not obvious to me; however, I should think it would be very large.

Tortoise: Achilles has a wonderful intuition for these things.

Crab: Yes. Well, as I was saying, Najunamar in his letter also proved that every even prime is the sum of two odd numbers, and that there are no solutions in positive integers to the equation

an + bn = cn

for n = 0.

Achilles: What? All these old classics of mathematics resolved in one fell swoop? He must be a genius of the first rank! Tortoise: But Achilles-aren't you even in the slightest skeptical?

Achilles: What? Oh, yes-skeptical. Well, of course I am. You don't think I believe that Mr. Crab got such a letter, do you? I don't fall for just anything, you know. So it must have been 5'ou, Mr. T, who received the letter!

Tortoise: Oh, no, Achilles, the part about Mr. C receiving the letter is quite true. What I meant was, aren't you skeptical about the content of the letter-its extravagant claims?

Achilles: Why should I be? Hmm ... Well, of course I am. I'm a very skeptical person, as both of you should well know by now. It's very hard to convince me of anything, no matter how true or false it is.

Tortoise: Very well put, Achilles. You certainly have a first-class awareness of your own mental workings.

Achilles: Did it ever occur to you, my friends, that these claims of Najunamar might be incorrect?

Crab: Frankly, Achilles, being rather conservative and orthodox myself, I was a bit concerned about that very point on first receiving the letter. In fact, I suspected at first that here was an out-and-out fraud. But on second thought, it occurred to me that not many types of people could manufacture such strange-sounding and complex results purely from their imagination. In fact, what it boiled down to was this question:

"Which is the more likely: a charlatan of such extraordinary ingenuity, or a mathematician of great genius?" And before long, I realized that the probabilities clearly favored the former.

Achilles: Didn't you directly checkout any of his amazing claims, however?

Crab: Why should I? The probability argument was the most convincing thing I had ever thought of; no mathematical proof would have equaled it. But Mr. T here insisted on rigor. I finally gave in to his insistence, and checked all of Najunamar's results. To my great surprise, each one of them was right. How he discovered them, I'll never know, however. He must have some amazing and inscrutable Oriental type of insight which we here in the Occident can have no inkling of. At present, that's the only theory which makes an sense to me.

Tortoise: Mr. Crab has always been a little more susceptible to mystical or fanciful explanations than I am. I have full confidence that whatever Najunamar did in his way has a complete parallel inside orthodox mathematics. There is no way of doing mathematics which is fundamentally different from what we now know, in my opinion.

Achilles: That is an interesting opinion. I suppose it has something to do with the Church-Turing Thesis and related topics.

Crab: Oh, well, let us leave these technical matters aside on such a fine day, and enjoy the quiet of the forest, the chirping of the birds, and the play of sunlight on the new leaves and buds. Ho!

Tortoise: I second the motion. After all, all generations of Tortoises have reveled in such delights of nature.