By
Douglas R. Hofstadter

GODEL, ESCHER,BACH:an Eternal Golden BraidBy Douglas R. Hofstadter

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• • • Ant Fugue

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. . . then, one by one, the four voices of the fugue chime in.)

Achilles: I know the rest of you won’t believe this, but the answer to the

question is staring us all in the face, hidden in the picture. It is simply

one word-but what an important one: "MU"!

CCrab: I know the rest of you won’t believe this, but the answer to the

question is staring us all in the face, hidden in the picture. It is simply

one word-but what an impvrtant one: "HOLISM"!

Achilles: Now hold on a minute. You must be seeing things. It’s plain as

day that the message of this picture is "MU", not "HOLISM"!

Crab: I beg your pardon, but my eyesight is extremely good. Please look

again, and then tell me if the the picture doesn’t say what I said it says!

Anteater: I know the rest of you won’t believe this, but the answer to the

question is staring us all in the face, hidden in the picture. It is simply

one word-but what an important one: "REDUCTIONISM"!

Crab: Now hold on a minute. You must be seeing things. It’s plain as day

that the message of this picture is "HOLISM", not "REDUCTION,SM"!

Achilles: Another deluded one! Not "HOLISM", not "REDUCTIONISM", but

"MU" is the message of this picture, and that much is certain.

Anteater: I beg your pardon, but my eyesight is extremely clear. Please

look again, and then see if the picture doesn’t say what I said it says.

Achilles: Don’t you see that the picture is composed of two pieces, and that

each of them is a single letter?

Crab: You are right about the two pieces, but you are wrong in your

identification of what they are. The piece on the left is entirely com-

posed of three copies of one word: "HOLISM"; and the piece on the right

is composed of many copies, in smaller letters, of the same word. Why

the letters are of different sizes in the two parts, I don’t know, but I

know what I see, and what I see is "HOLISM", plain as day. How you see

anything else is beyond me.

Antedter: You are right about the two pieces, but you are wrong in your

identification of what they are. The piece on the left is entirely com-

posed of many copies of one word: "REDUCTIONISM"; and the piece on

the right is composed of one single copy, in larger letters, of the same

word. Why the letters are of different sizes in the two parts, I don’t

know, but I know what I see, and what I see is "REDUCTIONISM", plain as

day. How you see anything else is beyond me.

Achilles: I know what is going on here. Each of you has seen letters which

compose, or are composed of, other letters. In the left-hand piece,

FIGURE 60. [Drawing by the authoT.]

there are indeed three "HOLISM’’’S, but each one of them is composed

out of smaller copies of the word ·’REDUCTIONISM". And in complemen-

tary fashion, in the right-hand piece, there is indeed one "REDUC-

TIONISM", but it is composed out of smaller copies of the word "HOLISM".

Now this is all fine and good, but in your silly squabble, the two of you

have actually missed the forest for the trees. You see, what good is it to

argue about whether "HOLISM" or "REDUCTIONISM" is right, when the

proper way to understand the matter is to transcend the question, by

answering "MU"?

Crab: I now see the picture as you have described it, Achilles, but I have

no idea of what you mean by the strange expression "transcending the

question".

Anteater: I now see the picture as you have described it, Achilles, but I

have no idea of what you mean by the strange expression "MU".

Achilles: I will be glad to indulge both of you, if you will first oblige me, by

telling me the meaning of thes.e strange expressions, "HOLISM" and

"REDUCTIONISM" .

Crab: HOLISM is the most natural thing in the world to grasp. It’s simply the

belief that "the whole is greater than the sum of its parts". No one in

his right mind could reject holism.

Anteater: REDUCTIONISM is the most natural thing in the world to grasp. It’s

simply the belief that "a whole can be understood completely if you

understand its parts, and the nature of their 'sum"’. No one in her

left brain could reject reductionism.

Crab: I reject reductionism. I challenge you to tell me, for instance, how to

understand a brain reductionistically. Any reductionistic explanation

of a brain will inevitably fall far short of explaining where the con-

sciousness experienced by a brain arises from.

Anteater: I reject holism. I challenge you to tell me, for instance, how a

holistic description of an ant colony sheds any more light on it than is

shed by a description of the ants inside it, and their roles, and their

interrelationships. Any holistic explanation of an ant colony will inevi-

tably fall far short of explaining where the consciousness experienced

by an ant colony arises from.

Achilles: Oh, no! The last thing which I wanted to do was to provoke

another argument. Anyway, now that I understand the controversy, I

believe that my explanation of "MU" will help greatly. You see, "MU" is

an ancient Zen answer which, when given to a question, UNASKS the

question. Here, the question seems to be, "Should the world be under-

stood via holism, or via reductionism?" And the answer of "MU" here

rejects the premises of the question, which are that one or the other

must be chosen. By unasking the question, it reveals a wider truth: that

there is a larger context into which both holistic and reductionistic

explanations fit.

Anteater: Absurd! Your "MU" is as silly as a cow’s moo. I’ll have none of this

Zen wishy-washiness.

Crab: Ridiculous! Your "MU" is as silly as a kitten’s mew. I’ll have none of

this Zen washy-wishiness.

Achilles: Oh, dear! We’re getting nowhere fast. Why have you stayed so

strangely silent, Mr. Tortoise? It makes me very uneasy. Surely you

must somehow be capable of helping straighten out this mess?

Tortoise: I know the rest of you won’t believe this, but the answer to the

question is staring us all in the face, hidden in the picture. It is simply

one word-but what an important one: HMU"!

aust as he says this, the fourth voice in the fugue being played enters,

exactly one octave below the first entry.)

Achilles: Oh, Mr. T, for once you have let me down. I was sure that you,

who always see the most deeply into things, would be able to resolve

this dilemma-but apparently, you have seen no further than I myself

saw. Oh, well, I guess I should feel pleased to have seen as far as Mr.

Tortoise, for once.

Tortoise: I beg your pardon, but my eyesight is. extremely fine. Please look

again, and then tell me if the picture doesn’t say what I said it says.

Achilles: But of course it does! You have merely repeated my own original

observation.

Tortoise: Perhaps "MU" exists in this picture on a deeper level than you

imagine, Achilles-an octave lower (figuratively speaking). But for

now I doubt that we can settle the dispute in the abstract. I would like

to see both the holistic and reductionistic points of view laid out more

explicitly; then there may be more of a basis for a decision. I would

very much like to hear a reductionistic description of an ant colony, for

instance.

Crab: Perhaps Dr. Anteater will tell you something of his experiences in

that regard. After all, he is by profession something of an expert on

that subject.

Tortoise: I am sure that we have much to learn from you, Dr. Anteater.

Could you tell us more about ant colonies, from a reductionistic point

of view?

Anteater: Gladly. As Mr. Crab mentioned to you, my profession has led me

quite a long way into the understanding of ant colonies.

Achilles: I can imagine! The profession of anteater would seem to be

synonymous with being an expert on ant colonies!

Anteater: I beg your pardon. "Anteater" is not my profession; it is my

species. By profession, I am a colony surgeon. I specialize in correcting

nervous disorders of the colony by the technique of surgical removal.

Achilles: Oh, I see. But what do you mean by "nervous disorders" of an ant

colony?

Anteater: Most of my clients suffer from some sort of speech impairment.

You know, colonies which have to grope for words in everyday situa-

tions. It can be quite tragic. I attempt to remedy the situation by,

uhh-removing’-the defective part of the colony. These operations

are sometimes quite involved, and of course years of study are re-

quired before one can perform them.

Achilles: But-isn’t it true that, before one can suffer from speech Im-

pairment, one must have the faculty of speech?

Anteater: Right.

Achilles: Since ant colonies don’t have that faculty, I am a little confused.

Crab: It’s too bad, Achilles, that you weren’t here last week, when Dr.

Anteater and Aunt Hillary were my house guests. I should have

thought of having you over then.

Achilles: Is Aunt Hillary your aunt, Mr. Crab?

Crab: Oh, no, she’s not really anybody’s aunt.

Anteater: But the poor dear insists that everybody should call her that,

even strangers. It’s just one of her many endearing quirks.

Crab: Yes, Aunt Hillary is quite eccentric, but such a merry old soul. It’s a

shame I didn’t have you over to meet her last week.

Anteater: She’s certainly one of the best-educated ant colonies I have ever

had the good fortune to know. The two of us have spent many a long

evening in conversation on the widest range of topics.

Achilles: I thought anteaters were devourers of ants, not patrons of ant-

'intellectualism!

Anteater: Well, of course the two are not mutually inconsistent. I am on

the best of terms with ant colonies. It’s just ANTS that I eat, not

colonies-and that is good for both parties: me, and the colony.

Achilles: How is it possible that-

Tortoise: How is it possible that-

Achilles: -having its ants eaten can do an ant colony any good?

Crab: How is it possible that-

Tortoise: -having a forest fire can do a forest any good?

Anteater: How is it possible that-

Crab: -having its branches pruned can do a tree any good?

Anteater: -having a haircut can do Achilles any good?

Tortoise: Probably the rest of you were too engrossed in the discussion to

notice the lovely stretto which just occurred in this Bach fugue.

Achilles: What is a stretto?

Tortoise: Oh, I’m sorry; I thought you knew the term. It is where one

theme repeatedly enters in one voice after another, with very little

delay between entries.

Achilles: If I listen to enough fugues, soon I’ll know all of these things and

will be able to pick them out myself, without their having to be pointed

out.

Tortoise: Pardon me, my friends. I am sorry to have interrupted. Dr.

Anteater was trying to explain how eating ants is perfectly consistent

with being a friend of an ant colony.

Achilles: Well, I can vaguely see how it might be possible for a limited and

regulated amount of ant consumption to improve the overall health of

a colony-but what is far more perplexing is all this talk about having

conversations with ant colonies. That’s impossible. An ant colony is

simply a bunch of individual ants running around at random looking

for food and making a nest.

Anteater: You could put it that way if you want to insist on seeing the trees

but missing the forest, Achilles. In fact, ant colonies, seen as wholes,

are quite well-defined units, with their own qualities, at times including

the mastery of language.

Achilles: I find it hard to imagine myself shouting something out loud in

the middle of the forest, and hearing an ant colony answer back.

Anteater: Silly fellow! That’s not the way it happens. Ant colonies don’t

converse out loud, but in writing. You know how ants form trails

leading them hither and thither?

Achilles: Oh, yes-usually straight through the kitchen sink and into my

peach jam.

Anteater: Actually, some trails contain information in coded form. If you

know the system, you can read what they’re saying just like a book.

Achilles: Remarkable. And can you communicate back to them?

Anteater: Without any trouble at all. That’s how Aunt Hillary and I have

conversations for hours. I take a stick and draw trails in the moist

ground, and watch the ants follow my trails. Presently, a new trail starts

getting formed somewhere. I greatly enjoy watching trails develop. As

they are forming, I anticipate how they will continue (and more often I

am wrong than right). When the trail is completed, I know what Aunt

Hillary is thinking, and I in turn make my reply.

Achilles: There must be some amazingly smart ants in that colony, I’ll say

that.

Anteater: I think you are still having some difficulty realizing the differ-

ence in levels here. Just as you would never confuse an individual tree

with a forest, so here you must not take an ant for the colony. You see,

all the ants in Aunt Hillary are as dumb as can be. They couldn’t

converse to save their little thoraxes!

Achilles: Well then, where does the ability to converse come from? It must

reside somewhere inside the colony! I don’t understand how the ants

can all be unintelligent, if Aunt Hillary can entertain you for hours

with witty banter.

Tortoise: It seems to me that the situation is not unlike the composition of

a human brain out of neurons. Certainly no one would insist that

individual brain cells have to be intelligent beings on their own, in

order to explain the fact that a person can have an intelligent conversa-

tion.

Achilles: Oh, no, clearly not. With brain cells, I see your point completely.

Only ... ants are a horse of another color. I mean, ants just roam

about at will, completely randomly, chancing now ahd then upon a

morsel of food ... They are free to do what they want to do, and with

that freedom, I don’t see at all how their behavior, seen as a whole, can

amount to anything coherent---{~specially something so coherent as the

brain behavior necessary for conversing.

Crab: It seems to me that the ants are free only within certain constraints.

For example, they are free to wander, to brush against each other, to

pick up small items, to work on trails, and so on. But they never step

out of that small world, that ant-system, which they are in. It would

never occur to them, for they don’t have the mentality to imagine

anything of the kind. Thus the ants are very reliable components, in

the sense that you can depend on them to perform certain kinds of

tasks in certain ways.

Achilles: But even so, within those limits they are still free, and they just act

at random, running about incoherently without any regard for the

thought mechanisms of a higher-level being which Dr. Anteater asserts

they are merely components of.

Anteater: Ah, but you fail to recognize one thing, Achilles-the regularity

of statistics.

Achilles: How is that?

Anteater: For example, even though ants as individuals wander about in

what seems a random way, there are nevertheless overall trends, in-

volving large numbers of ants, which can emerge from that chaos.

Achilles: Oh, I know what you mean. In fact, ant trails are a perfect

example of such a phenomenon. There, you have really quite unpre-

dictable motion on the part of any single ant-and yet, the trail itself

seems to remain well-defined and stable. Certainly that must mean that

the individual ants are not just running about totally at random.

Anteater: Exactly, Achilles. There is some degree of communication

among the ants, just enough to keep them from wandering off com-

pletely at random. By this minimal communication they can remind

each other that they are not alone but are cooperating with teammates.

It takes a large number of ants, all reinforcing each other this way, to

sustain any activity-such as trail-building-for any length of time.

Now my very hazy understanding of the operation of brains leads me

to believe that something similar pertains to the firing of neurons. Isn’t

it true, Mr. Crab, that it takes a group of neurons firing in order to

make another neuron fire?

Crab: Definitely. Take the neurons in Achilles' brain, for example. Each

neuron receives signals from neurons attached to its input lines, and if

the sum total of inputs at any moment exceeds a critical threshold,

then that neuron will fire and send its own output pulse rushing off to

other neurons, which may in turn fire-and on down the line it goes.

The neural flash swoops relentlessly in its Achillean path, in shapes

stranger then the dash of a gnatchungry swallow; every twist, every

turn foreordained by the neural structure in Achilles' brain, until

sensory input messages interfere.

Achilles: Normally, I think that I’M in control of what I think-but the way

you put it turns it all inside out, so that it sounds as though "I" am just

what comes out of all this neural structure, and natural law. It makes

what I consider my SELF sound at best like a by-product of an organism

governed by natural law , and at worst, an artificial notion produced by

my distorted perspective. In other words, you make me feel like I don’t

know who-or what-I am, if anything.

Tortoise: You’ll come to understand much better as we go along. But Dr.

Anteater-what do you make of this similarity?

Anteater: I knew there was something parallel going on in the two very

different systems. Now I understand it much better. It seems that

group phenomena which have coherence-trail-building, for

example-will take place only when a certain threshold number of ants

get involved. If an effort is initiated, perhaps at random, by a few ants

in some locale, one of two things can happen: either it will fizzle out

after a brief sputtering start-

Achilles: When there aren’t enough ants to keep the thing rolling?

Anteater: Exactly. The other thing that can happen is that a critical mass of

ants is present, and the thing will snowball, bringing more and more

ants into the picture. In the latter case, a whole "team" is brought into

being which works on a single project. That project might be trail-

making, or food-gathering, or it might involve nest-keeping. Despite

the extreme simplicity of this scheme on a small scale, it can give rise to

very complex consequences on a larger scale.

Achilles: I can grasp the general idea of order emerging from chaos, as

you sketch it, but that still is a lpng way from the ability to converse.

After all, order also emerges from chaos when molecules of a gas

bounce against each other randomly-yet all that results there is an

amorphous mass with but three parameters to characterize it: volume,

pressure, and temperature. Now that’s a far cry from the ability to

understand the world, or to talk about it!

Anteater: That highlights a very interesting difference between the expla-

nation of the behavior of an ant colony and the explanation of the

behavior of gas inside a container. One can explain the behavior of the

gas simply by calculating the statistical properties of the motions of its

molecules. There is no need to discuss any higher elements of struc-

ture than molecules, except the full gas itself. On the other hand, in an

ant colony, you can’t even begin to understand the activities of the

colony unless you go through several layers of structure.

Achilles: I see what you mean. In a gas, one jump takes you from the

lowest level-molecules-to the highest level-the full gas. There are

no intermediate levels of organization. Now how do intermediate

levels of organized activity arise in an ant colony?

Anteater: It has to do with the existence of several different varieties of

ants inside any colony.

Achilles: Oh, yes. I think I have heard about that. They are called "castes",

aren’t they?

Anteater: That’s correct. Aside from the queen, there are males, who do

practically nothing towards the upkeep of the nest, and then-

Achilles: And of course there are soldiers-Glorious Fighters Against

Communism!

Crab: Hmm ... I hardly think that could be right, Achilles. An ant colony

is quite communistic internally, so why would its soldiers fight against

communism? Or am I right, Dr. Anteater?

Anteater: Yes, about colonies you are right, Mr. Crab; they are indeed

based on somewhat communistic principles. But about soldiers Achil-

les is somewhat naIve. In fact, the so-called "soldiers" are hardly adept

at fighting at all. They are slow, ungainly ants with giant heads, who

can snap with their strong jaws, but are hardly to be glorified. As in a

true communistic state, it is rather the workers who are to be glorified.

It is they who do most of the chores, such as food-gathering, hunting,

and nursing of the young. It is even they who do most of the fighting.

Achilles: Bah. That is an absurd state of affairs. Soldiers who won’t fight!

Anteater: Well, as I just said, they really aren’t soldiers at all. It’s the

workers who are soldiers; the soldiers are just lazy fatheads.

Achilles: Oh, how disgraceful! Why, if I were an ant, I’d put some disci-

pline in their ranks! I’d knock some sense into those fatheads!

Tortoise: If you were an ant? How could you be an ant? There is no way to

map your brain onto an ant brain, so it seems to me to be a pretty

fruitless question to worry over. More reasonable would be the propo-

sition of mapping your brain onto an ant colony ... But let us not get

sidetracked. Let Dr. Anteater continue with his most illuminating

description of castes and their role in the higher levels of organization.

Anteater: Very well. There are all sorts of tasks which must be ac-

complished in a colony, and individual ants develop specializations.

Usually an ant’s specialization changes as the ant ages. And of course it

is also dependent on the ant’s caste. At anyone moment, in any small

area of a colony, there are ants of all types present. Of course, one

caste may be be very sparse in some places and very dense in others.

Crab: Is the density of a given caste, or specialization, just a random

thing? Or is there a reason why ants of one type might be more heavily

concentrated in certain areas, and less heavily in others?

Anteater: I’m glad you brought that up, since it is of crucial importance in

understanding how a colony thinks. In fact, there evolves, over a long

period of time, a very delicate distribution of castes inside a colony.

And it is this distribution which allows the colony to have the complex-

ity which underlies the ability to converse with me.

Achilles: It would seem to me that the constant motion of ants to and fro

would completely prevent the possibility of a very delicate distribution.

Any delicate distribution would be quickly destroyed by all the random

motions of ants, just as any delicate pattern among molecules in a gas

would not survive for an instant, due to the random bombardment

from all sides.

Anteater: In an ant colony, the situation is quite the contrary. In fact, it is

just exactly the constant to-ing and fro-ing of ants inside the colony

which adapts the caste distribution to varying situations, and thereby

preserves the delicate caste distribution. You see, the caste distribution

cannot remain as one single rigid pattern; rather, it must constantly be

changing so as to reflect, in some manner, the real-world situation with

which the colony is dealing, and it is precisely the motion inside the

colony which updates the caste distribution, so as to keep it in line with

the present circumstances facing the colony.

Tortoise: Could you give an example?

Anteater: Gladly. When I, an anteater, arrive to pay a visit to Aunt Hillary,

all the foolish ants, upon sniffing my odor, go into a panic-which

means, of course, that they begin running around completely diffe-

rently from the way they were before I arrived.

Achilles: But that’s understandable, since you’re a dreaded enemy of the

colony.

Anteater: Oh, no. I must reiterate that, far from being an enemy of the

colony, I am Aunt Hillary’s favorite companion. And Aunt Hillary is

my favorite aunt. I grant you, I’m quite feared by all the individual

ants in the colony-but that’s another matter entirely. In any case, you

see that the ants' action in response to my arrival completely changes

the internal distribution of ants.

Achilles: That’s clear.

Anteater: And that sort of thing is the updating which I spoke of. The new

distribution reflects my presence. One can describe the change from

old state to new as having added a "piece of knowledge" to the colony.

Achilles: How can you refer to the distribution of different types of ants

inside a colony as a "piece of knowledge"?

Anteater: Now there’s a vital point. It requires some elaboration. You see,

what it comes down to is how you choose to describe the caste distribu-

tion. If you continue to think in terms of the lower levels-individual

ants-then you miss the forest for the trees. That’sjust too microscopic

a level, and when you think microscopically, you’re bound to miss some

large-scale features. You’ve got to find the proper high-level

framework in which to describe the caste distribution-only then will it

make sense how the caste distribution can encode many pieces of

knowledge.

Achilles: Well, how DO you find the proper-sized units in which to describe

the present state of the colony, then?

Anteater: All right. Let’s begin at the bottom. When ants need to get

something done, they form little "teams", which stick together to

perform a chore. As I mentioned earlier, small groups of ants are

constantly forming and unforming. Those which actually exist for a

while are the teams, and the reason they don’t fall apart is that there

really is something for them to do.

Achilles: Earlier you said that a group will stick together if its size exceeds a

certain threshold. Now you’re saying that a group will stick together if

there is something for it to do.

Anteater: They are equivalent statements. For instance, in food-gathering,

if there is an inconsequential amount of food somewhere which gets

discovered by some wandering ant who then attempts to communicate

its enthusiasm to other ants, the number of ants who respond will be

proportional to the size of the food sample-and an inconsequential

amount will not attract enough ants to surpass the threshold. Which is

exactly what I meant by saying there is nothing to do-too little food

ought to be ignored.

Achilles: I see. I assume that these "teams" are one of the levels of struc-

ture falling somewhere in between the single-ant level and the colony

level.

Anteater: Precisely. There exists a special kind of team, which I call a

"signal"-and all the higher levels of structure are based on signals. In

fact, all the higher entities are collections of signals acting in concert.

There are teams on higher levels whose members are not ants, but

teams on lower levels. Eventually you reach the lowest-level teams-

which is to say, signals-and below them, ants.

Achilles: Why do signals deserve their suggestive name?

Anteater: It comes from their function. The effect of signals is to transport

ants of various specializations to appropriate parts of the colony. So the

typical story of a signal is thus; it (omes into existence by exceeding the

threshold needed for survival, then it migrates for some distance

through the colony, and at some point it more or less disintegrates into

its individual members, leaving them on their own.

Achilles: It sounds like a wave, carrying sand dollars and seaweed from

afar, and leaving them strewn, high and dry, on the shore.

Anteater: In a way that’s analogous, since the team does indeed deposit

something which it has carried from a distance, but whereas the water

in the wave rolls back to the sea, there is no analogous carrier substance

in the case of a signal, since the ants themselves compose it.

Tortoise: And I suppose that a signal loses its coherency just at some spot

in the colony where ants of that type were needed in the first place.

Anteater: Naturally.

Achilles: Naturally? It’s not so obvious to ME that a signal should always go

just where it is needed. And even if it goes in the right direction, how

does it figure out where to decompose? How does it know it has

arrived?

Anteater: Those are extremely important matters, since they involve ex-

plaining the existence of purposeful behavior-or what seems to be

purposeful behavior-on the part of signals. From the description, one

would be inclined to characterize the signals' behavior as being

oriented towards filling a need, and to call it "purposeful". But you can

look at it otherwise.

Achilles: Oh, wait. Either the behavior IS purposeful, or it is NOT. I don’t

see how you can have it both ways.

Anteater: Let me explain my way of seeing things, and then see if you

agree. Once a signal is formed, there is no awareness on its part that it

should head off in any particular direction. But here, the delicate caste

distribution plays a crucial role. It is what determines the motion of

signals through the colony, and also how long a signal will remain

stable, and where it will "dissolve".

Achilles: So everything depends on the caste distribution, eh?

Anteater: Right. Let’s say a signal is moving along. As it goes, the ants

which compose it interact, either by direct contact or by exchange of

scents, with ants of the local neighborhoods which it passes through.

The contacts and scents provide information about local matters of

urgency, such as nest-building, or nursing, or whatever. The signal will

remain glued together as long as the local needs are different from

what it can supply; but if it CAN contribute, it disintegrates, spilling a

fresh team of usable ants onto the scene. Do you see now how the caste

distribution acts as an overall guide of the teams inside the colony?

Achilles: I do see that.

Anteater: And do you see how this way of looking at things requires

attributing no sense of purpose to the signal?

Achilles: I think so. Actually, I’m beginning to see things from two differ-

ent vantage points. From an ant’s-eye point of view, a signal has NO

purpose. The typical ant in a signal is just meandering around the

colony, in search of nothing in particular, until it finds that it feels like

stopping. Its teammates usually agree, and at that moment the team

unloads itself by crumbling apart, leaving just its members but none of

its coherency. No planning is required, no looking ahead; nor is any

search required, to determine the proper direction. But from the

COLONY’S point of view, the team has just responded to a message

which was written in the language of the caste distribution. Now from

this perspective, it looks very much like purposeful activity.

Crab: What would happen if the caste distribution were entirely random?

Would signals still band and disband?

Anteater: Certainly. But the colony would not last long, due to the

meaninglessness of the caste distribution.

Crab: Precisely the point I wanted to make. Colonies survive because their

caste distribution has meaning, and that meaning is a holistic aspect,

invisible on lower levels. You lose explanatory power unless you take

that higher level into account.

Anteater: I see your side; but I believe you see things too narrowly.

Crab: How so?

Anteater: Ant colonies have been subjected to the rigors of evolution for

billions of years. A few mechanisms were selected for, and most were

selected against. The end result was a set of mechanisms which make

ant colonies work as we have been describing. If you could watch the

whole process in a movie-running a billion or so times faster than life,

of course-the emergence of various mechanisms would be seen as

natural responses to external pressures,just as bubbles in boiling water

are natural responses to an external heat source. I don’t suppose you

see "meaning" and "purpose" in the bubbles in boiling water-or do

you?

Crab: No, but-

Anteater: Now that’s MY point. No matter how big a bubble is, it owes its

existence to processes on the molecular level, and you can forget about

any "higher-level laws". The same goes for ant colonies and their

teams. By looking at things from the vast perspective of evolution, you

can drain the whole colony of meaning and purpose. They become

superfluous notions.

Achilles: Why, then, Dr. Anteater, did you tell me that you talked with

Aunt Hillary? It now seems that you would deny that she can talk or

think at all.

Anteater: I am not being inconsistent, Achilles. You see, I have as much

difficulty as anyone else in seeing things on such a grandiose time scale,

so I find it much easier to change points of view. When I do so,

forgetting about evolution and seeing things in the here and now, the

vocabulary of teleology comes back: the MEANING of the caste distribu-

tion and the PURPOSEFULNESS of signals. This not only happens when I

think of ant colonies, but also when I think about my own brain and

other brains. However, with some effort I can always remember the

other point of view if necessary, and drain all these systems of mean-

ing, too.

Crab: Evolution certainly works some miracles. You never know the next

trick it will pull out of its sleeve. f’or instance, it wouldn’t surprise me

one bit if it were theoretically possible for two or more "signals" to pass

through each other, each one unaware that the other one is also a

signal; each one treating the other as if it were just part of the

background population.

Anteater: It is better than theoretically possible; in fact it happens

routinely!

Achilles: Hmm ... What a strange image that conjures up in my mind. I

can just imagine ants moving in four different directions, some black,

some white, criss-crossing, together forming an orderly pattern, almost

like-like-

Tortoise: A fugue, perhaps?

Achilles: Yes-that’s it! An ant fugue!

Crab: An interesting image, Achilles. By the way, all that talk of boiling

water made me think of tea. Who would like some more?

Achilles: I could do with another cup, Mr. C.

Crab: Very good.

Achilles: Do you suppose one could separate out the different visual

"voices" of such an "ant fugue"? I know how hard it is for me-

Tortoise: Not for me, thank you.

Achilles: -to track a single voice-

Anteater: I’d like some, too, Mr. Crab-

Achilles: -in a musical fugue

Anteater: -if it isn’t too much trouble.

Achilles: -when all of them-

Crab: Not at all. Four cups of tea-

Tortoise: Three!

Achilles: -are going at once.

Crab: -coming right up!

Anteater: That’s an interesting thought, Achilles. But it’s unlikely that

anyone could draw such a picture in a convincing way.

Achilles: That’s too bad.

Tortoise: Perhaps you could answer this, Dr. Anteater. Does a signal, from

its creation until its dissolution, always consist of the same set of ants?

Anteater: As a matter of fact, the individuals in a signal sometimes break

off and get replaced by others of the same caste, if there are a few in

the area. Most often, s.ignals arrive at their disintegration points with

nary an ant in common with their starting lineup.

Crab: I can see that the signals are constantly affecting the caste distribu-

tion throughout the colony, and are doing so in response to the

internal needs of the colony-which in turn reflect the external situa-

tion which the colony is faced with. Therefore the caste distribution, as

you said, Dr. Anteater, gets continually updated in a way which ulti-

mately reflects the outer world.

Achilles: But what about those intermediate levels of structure? You were

saying that the caste distribution should best be pictured not in terms

of ants or signals, but in terms of teams whose members were other

teams, whose members were other teams, and so on until you come

down to the ant level. And you said that that was the key to under-

standing how it was possible to describe the caste distribution as encod-

ing pieces of information about the world.

Anteater: Yes, we are coming to all that. I prefer to give teams of a

sufficiently high level the name of "symbols". Mind you, this sense of

the word has some significant differences from the usual sense. My

"symbols" are ACTIVE SUBSYSTEMS of a complex system, and they are

composed of lower-level active subsystems ... They are therefore

quite different from PASSIVE symbols, external to the system, such as

letters of the alphabet or musical notes, which sit there immobile,

waiting for an active system to process them.

Achilles: Oh, this is rather complicated, isn’t it? I just had no idea that ant

colonies had such an abstract structure.

Anteater: Yes, it’s quite remarkable. But all these layers of structure are

necessary for the storage of the kinds of knowledge which enable an

organism to be "intelligent" in any reasonable sense of the word. Any

system which has a mastery of language has essentially the same under-

lying sets of levels.

Achilles: Now just a cotton-picking minute. Are you insinuating that my

brain consists of, at bottom, just a bunch of ants running around?

Anteater: Oh, hardly. You took me a little too literally. The lowest level

may be utterly different. Indeed, the brains of anteaters, for instance,

are not composed of ants. But when you go up a level or two in a brain,

you reach a level whose elements have exact counterparts in other

systems of equal intellectual strength-such as ant colonies.

Tortoise: That is why it would be reasonable to think of mapping your

brain, Achilles, onto an ant colony, but not onto the brain of a mere

ant.

Achilles: I appreciate the compliment. But how would such a mapping be

carried out? For instance, what in my brain corresponds to the low-

level teams which you call signals?

Anteater: Oh, I but dabble in brains, and therefore couldn’t set up the map

in its glorious detail. But-and correct me if I’m wrong, Mr. Crab-I

would surmise that the brain counterpart to an ant colony’s signal is

the firing of a neuron; or perhaps it is a larger-scale event, such as a

pattern of neural firings.

Crab: I would tend to agree. But don’t you think that, for the purposes of

our discussion, delineating the exact counterpart is not in itself crucial,

desirable though it might be? It seems to me that the main idea is that

such a correspondence does exist, even if we don’t know exactly how to

define it right now. I would only question one point, Dr. Anteater,

which you raised, and that concerns the level at which one can have

faith that the correspondence begins. You seemed to think that a

SIGNAL might have a direct counterpart in a brain; whereas I feel that

it is only at the level of your ACTIVE SYMBOLS and above that it is likely

that a correspondence must exist.

Anteater: Your interpretation may very well be more accurate than mine,

Mr. Crab. Thank you for bringing out that subtle point.

Achilles: What does a symbol do that a signal couldn’t do?

Anteater: It is something like the difference between words and letters.

Words, which are meaning-carrying entities, are composed of letters,

which in themselves carry no meaning. This gives a good idea of the

difference between symbols and signals. In fact it is a useful analogy, as

long as you keep in mind the fact that words and letters are PASSIVE,

symbols and signals are ACTIVE.

Achilles: I’ll do so, but I’m not sure I understand why it is so vital to stress

the difference between active and passive entities.

Anteater: The reason is that the meaning which you attribute to any

passive symbol, such as a word on a page, actually derives from the

meaning which is carried by corresponding active symbols in your

brain. So that the meaning of passive symbols can only be properly

understood when it is related to the meaning of active symbols.

Achilles: All right. But what is it that endows a SYMBOL-an active one, to

be sure-with meaning, when you say that a SIGNAL, which is a per-

fectly good entity in its own right, has none?

Anteater: It all has to do with the way that symbols can cause other symbols

to be triggered. When one symbol becomes active, it does not do so in

isolation. It is floating about, indeed, in a medium, which is charac-

terized by its caste distribution.

Crab: Of course, in a brain there is no such thing as a caste distribution,

but the counterpart is the "brain state". There, you describe the states

of all the neurons, and all the interconnections, and the threshold for

firing of each neuron.

Anteater: Very well; let’s lump "caste distribution" and "brain state" under

a common heading, and call them just the "state". Now the state can be

described on a low level or on a high level. A low-level description of

the state of an ant colony would involve painfully specifying the loca-

tion of each ant, its age and caste, and other similar items. A very

detailed description, yielding practically no global insight as to WHY it

is in that state. On the other hand, a description on a high level would

involve specifying which symbols could be triggered by which combi-

nations of other symbols, under what conditions, and so forth.

Achilles: What about a description on the level of signals, or teams?

Anteater: A description on that level would fall somewhere in between the

low-level and symbol-level descriptions. It would contain a great deal

of information about what is actually going on in specific locations

throughout the colony, although certainly less than an ant-by-ant de-

scription, since teams consist of clumps of ants. A team-by-team de-

scription is like a summary of an ant-by-ant description. However, you

have to add extra things which were not present in the ant-by-ant

description-such as the relationships between teams, and the supply

of various castes here and there. This extra complication is the price

you pay for the right to summarize.

Achilles: It is interesting to me to compare the merits of the descriptions at

various levels. The highest-level description seems to carry the most

explanatory power, in that it gives you the most intuitive picture of the

ant colony, although strangely enough, it leaves out seemingly the

most important feature-the ants.

Anteater: But you see, despite appearances, the ants are not the most

important feature. Admittedly, were it not for them, the colony

wouldn’t exist; but something equivalent-a brain-can exist, ant-free.

So, at least from a high-level point of view, the ants are dispensable.

Achilles: I’m sure no ant would embrace your theory with eagerness.

Anteater: Well, I never met an ant with a high-level point of view.

Crab: What a counterintuitive picture you paint, Dr. Anteater. It seems

that, if what you say is true, in order to grasp the whole structure, you

have to describe it omitting any mention of its fundamental building

blocks.

Anteater: Perhaps I can make it a little clearer by an analogy. Imagine you

have before you a Charles Dick.ens novel.

Achilles: The Pickwick Papers-will that do?

Anteater: Excellently! And now imagine trying the following game: you

must find a way of mapping letters onto ideas, so that the entire

Pickwick Papers makes sense when you read it letter by letter.

Achilles: Hmm ... You mean that every time I hit a word such as "the", I

have to think of three definite concepts, one after another, with no

room for variation?

Anteater: Exactly. They are the 't’-concept, the 'h' -concept, and the

'e' -concept-and every time, those concepts are as they were the pre-

ceding time.

Achilles: Well, it sounds like that would turn the experience of "reading"

The Pickwick Papers into an indescribably boring nightmare. It would be

an exercise in meaninglessness, no matter what concept I associated

with each letter.

Anteater: Exactly. There is no natural mapping from the individual letters

into the real world. The natural mapping occurs on a higher level-

between words, and parts of the real world. If you wanted to describe

the book, therefore, you would make no mention of the letter level.

Achilles: Of course not! I’d describe the plot and the characters, and so

forth.

Anteater: So there you are. You would omit all mention of the building

blocks, even though the book exists thanks to them. They are the

medium, but not the message.

Achilles: All right-but what about ant colonies?

Anteater: Here, there are active signals instead of passive letters, and

active symbols instead of passive words-but the idea carries over.

Achilles: Do you mean I couldn’t establish a mapping between signals and

things in the real world?

Anteater: You would find that you could not do it in such a way that the

triggering of new signals would make any sense. Nor could you suc-

ceed on any lower level-for example the ant level. Only on the symbol

level do the triggering patterns make sense. Imagine, for instance, that

one day you were watching Aunt Hillary when I arrived to pay a call.

You could watch as carefully as you wanted, and yet you would proba-

bly perceive nothing more than a rearrangement of ants.

Achilles: I’m sure that’s accurate.

Anteater: And yet, as I watched, reading the higher level instead of the

lower level, I would see several dormant symbols being awakened,

those which translate into the thought, "Oh, here’s that charming Dr.

Anteater again-how pleasant!"-or words to that effect.

Achilles: That sounds like what happened when the four of us all

found different levels to read in the MU-picture-or at least THREE of

us did ...

Tortoise: What an astonishing coincidence that there should be such a

resemblance between that strange picture which I chanced upon in the

Well-Tempered Clavier, and the trend of our conversation.

Achilles: Do you think it’s just coincidence?

Tortoise: Of course.

Anteater: Well, I hope you can grasp now how the thoughts in Aunt

Hillary emerge from the manipulation of symbols composed of signals

composed of teams composed of lower-level teams, all the way down to

ants.

Achilles: Why do you call it "symbol manipulation"? Who does the manip-

ulating, if the symbols are themselves active? Who is the agent?

Anteater: This gets back to the question which you earlier raised about

purpose. You’re right that symbols themselves are active, but the

activities which they follow are nevertheless not absolutely free. The

activities of all symbols are strictly determined by the state of the full

system in which they reside. Therefore, the full system is responsible

for how its symbols trigger each other, and so it is quite reasonable to

speak of the full system as the "agent". As the symbols operate, the

state of the system gets slowly transformed, or updated. But there are

many features which remain over time. It is this partially constant,

partially varying system which is the agent. One can give a name to the

full system. For example, Aunt Hillary is the "who" who can be said to

manipulate her symbols; and you are similar, Achilles.

Achilles: That’s quite a strange characterization of the notion of who I am.

I’m not sure I can fully understand it, but I will give it some thought.

Tortoise: It would be quite interesting to follow the symbols in your brain

as you do that thinking about the symbols in your brain.

Achilles: That’s too complicated for me. I have trouble enough just trying

to picture how it is possible to look at an ant colony and read it on the

symbol level. I can certainly imagine perceiving it at the ant level; and

with a little trouble, I can imagine what it must be like to perceive it at

the signal level; but what in the world can it be like to perceive an ant

colony at the symbol level?

Anteater: One only learns through long practice. But when one is at my

stage, one reads the top level of an ant colony as easily as you yourself

read the "MU" in the MU-picture.

Achilles: Really? That must be an amazing experience.

Anteater: In a way-but it is also one which is quite familiar to you,

Achilles.

Achilles: Familiar to me? What do you mean? I have never looked at an ant

colony on anything but the ant level.

Anteater: Maybe not; but ant colonies are no different from brains m

many respects.

Achilles: I have never seen nor read any brain either, however.

Anteater: What about your OWN brain? Aren’t you aware of your own

thoughts? Isn’t that the essence of consciousness? What else are you

doing but reading your own brain directly at the symbol level?

Achilles: I never thought of it that way. You mean that I bypass all the

lower levels, and only see the topmost level?

Anteater: That’s the way it is, with conscious systems. They perceive them-

selves on the symbol level only, and have no awareness of the lower

levels, such as the signal levels.

Achilles: Does it follow that in a brain, there are active symbols which are

constantly updating themselves so that they reRect the overall state of

the brain itself, always on the symbol level?

Anteater: Certainly. In any conscious system there are symbols which

represent the brain state, and they are themselves part of the very

brain state which they symbolize. For consciousness requires a large

degree of self-consciousness.

Achilles: That is a weird notion. It means that although there is frantic

activity occurring in my brain at all times, I am only capable of register-

ing that activity in one way-on the symbol level; and I am completely

insensitive to the lower levels. It is like being able to read a Dickens

novel by direct visual perception, without ever having learned the

letters of the alphabet. I can’t imagine anything as weird as that really

happening.

Crab: But precisely that sort of thing DID happen when you read "MU",

without perceiving the lower levels "HOLISM" and "REDUCTIONISM".

Achilles: You’re right-I bypassed the lower levels, and saw only the top. I

wonder if I’m missing all sorts of meaning on lower levels of my brain

as well, by reading only the symbol level. It’s too bad that the top level

doesn’t contain all the information about the bottom level, so that by

reading the top, one also learns what the bottom level says. But I guess

it would be naive to hope that the top level encodes anything from the

bottom level-it probably doesn’t percolate up. The MU-picture is the

most striking possible example of that: there, the topmost level says

only "MU", which bears no relation whatever to the lower levels!

Crab: That’s absolutely true. (Picks up the MU-picture, to inspect it more

closely.) Hmm ... There’s something strange about the smallest letters

in this picture; they’re very wiggly ...

Anteater: Let me take a look. (Peers closely at the MU-picture.) I think there’s

yet another level, which all of us missed!

Tortoise: Speak for yourself, Dr. Anteater.

Achilles: Oh, no-that can’t be! Let me see. (Looks very carefully.) I know the

rest of you won’t believe this, but the message of this picture is staring

us all in the face, hidden in its depths. It is simply one word, repeated

over and over again, like a mantra-but what an important one: "MU"!

What do you know! It is the same as the top level! And none of us

suspected it in the least.

Crab: We would never have noticed it if it hadn’t been for you, Achilles.

Anteater: I wonder if the coincidence of the highest and lowest levels

happened by chance? Or was it a purposeful act carried out by some

creator?

Crab: How could one ever decide that?

Tortoise: I don’t see any way to do so, since we have no idea why that

particular picture is in the Crab’s edition of the Well-Tempered Clavier.

Anteater: Although we have been having a lively discussion, I have still

managed to listen with a good fraction of an ear to this very long and

complex four-voice fugue. It is extraordinarily beautiful.

Tortoise: It certainly is. And now, injust a moment, comes an organ point.

Achilles: Isn’t an organ point what happens when a piece of music slows

down slightly, settles for a moment or two on a single note or chord,

and then resumes at normal speed after a short silence?

Tortoise: No, you’re thinking of a "fermata"-a sort of musical semicolon.

Did you notice there was one of those in the prelude?

Achilles: I guess I must have missed it.

Tortoise: Well, you have another chance coming up to hear a fermata-in

fact, there are a couple of them coming up, towards the end of this

fugue.

Achilles: Oh, good. You’ll point them out in advance, won’t you?

Tortoise: If you like.

Achilles: But do tell me, what is an organ point?

Tortoise: An organ point is the sustaining of a single note by one of the

voices in a polyphonic piece (often the lowest voice), while the other

voices continue their own independent lines. This organ point is on the

note of G. Listen carefully, and you’ll hear it.

Anteater: There occurred an incident one day when I visited with Aunt

Hillary which reminds me of your suggestion of observing the symbols

in Achilles' brain as they create thoughts which are about themselves.

Crab: Do tell us about it.

Anteater: Aunt Hillary had been feeling very lonely, and was very happy

to have someone to talk to that day. So she gratefully told me to help

myself to the juiciest ants I could find. (She’s always been most gener-

ous with her ants.)

Achilles: Gee!

Anteater: It just happened that I had been watching the symbols which

were carrying out her thoughts, because in them were some particu-

larly juicy-looking ants.

Achilles: Gee!

Anteater: So I helped myself to a few of the fattest ants which had been

parts of the higher-level symbols which I had been reading. Specifical-

ly, the symbols which they were part of were the ones which had

expressed the thought, "Help yourself to any of the ants which look

appetizing ."

Achilles: Gee!

Anteater: Unfortunately for them, but fortunately for me, the little bugs

didn’t have the slightest inkling of what they were collectively telling

me, on the symbol level.

Achilles: Gee! That is an amazing wraparound. They were completely

unconscious of what they were participating in. Their acts could be

seen as part of a pattern on a higher level, but of course they were

completely unaware of that. Ah, what a pity-a supreme irony, III

fact-that they missed it.

Crab: You are right, Mr. T -that was a lovely organ point.

Anteater: I had never heard one before, but that one was so conspicuous

that no one could miss it. Very effective.

Achilles: What? Has the organ point already occurred? How can I not

have noticed it, if it was so blatant?

Tortoise: Perhaps you were so wrapped up in what you were saying that

you were completely unaware of it. Ah, what a pity-a supreme irony,

in fact-that you missed it.

Crab: Tell me, does Aunt Hillary live in an anthill?

Anteater: Well, she owns a rather large piece of property. It used to belong

to someone else, but that is rather a sad story. In any case, her estate is

quite expansive. She lives rather sumptuously, compared to many

other colonies.

Achilles: How does that jibe with the communistic nature of ant colonies

which you earlier described to us? It sounds quite inconsistent, to me,

to preach communism and to live in a fancy estate!

Anteater: The communism is on the ant level. In an ant colony all ants

work for the common good, even to their own individual detriment at

times. Now this is simply a built-in aspect of Aunt Hillary’s structure,

but for all I know, she may not even be aware of this internal com-

munism. Most human beings are not aware of anything about their

neurons; in fact they probably are quite content not to know anything

about their brains, being somewhat squeamish creatures. Aunt Hillary

is also somewhat squeamish; she gets rather antsy whenever she starts

to think about ants at all. So she avoids thinking about them whenever

possible. I truly doubt that she knows anything about the communistic

society which is built into her very structure. She herself is a staunch

believer in libertarianism-you know, laissez-faire and all that. So it

makes perfect sense, to me at least, that she should live in a rather

sumptuous manor.

Tortoise: As I turned the page just now, while following along in this lovely

edition of the Well-Tempered Clavier, I noticed that the first of the two

fermatas is coming up soon-so you might listen for it, Achilles.

Achilles: I will, I will.

Tortoise: Also, there’s a most curious picture facing this page.

Crab: Another one? What next?

Tortoise: See for yourself. (Passes the score over to the Crab.)

Crab: Aha! It’s just a few bunches of letters. Let’s see-there are various

numbers of the letters ']’, 'S’, 'B’, 'm’, 'a’, and 't’. It’s strange, how the

first three letters grow, and then the last three letters shrink again.

Anteater: May I see it?

Crab: Why, certainly.

Anteater: Oh, by concentrating on details, you have utterly missed the big

picture. In reality, this group of letters is 'f’, 'e’, 'r’, 'A’, 'C’, 'H’, without

any repetitions. First they get smaller, then they get bigger. Here,

Achilles-what do you make of it?

Achilles: Let me see. Hmm. Well, I see it as a set of upper-case letters

which grow as you move to the right.

Tortoise: Do they spell anything?

Achilles: Ah ... "J. S. BACH". Oh! I understand now. It’s Bach’s name!

Tortoise: Strange that you should see it that way. I see it as a set of

lower-case letters, shrinking as they move to the right, and ... spelling

out ... the name of ... (Slows down slightly, especially drawing out the last

few words. Then there is a brief silence. Suddenly he resumes as if nothing

unusual had happened.) -"fermat".

Achilles: Oh, you’ve got Fermat on the brain, I do believe. You see Fer-

mat’s Last Theorem everywhere.

Anteater: You were right, Mr. Tortoise-I just heard a charming little

fermata in the fugue.

Crab: So did I.

Achilles: Do you mean everybody heard it but me? I’m beginning to feel

stupid.

Tortoise: There, there, Achilles-don’t feel bad. I’m sure you won’t miss

Fugue’s Last Fermata (which is coming up quite soon). But, to return

to our previous topic, Dr. Anteater, what is the very sad story which

you alluded to, concerning the former owner of Aunt Hillary’s prop-

erty?

Anteater: The former owner was an extraordinary individual, one of the

most creative ant colonies who ever lived. His name was Johant Sebas-

tiant Fermant, and he was a mathematiciant by vocation, but a

musiciant by avocation.

Achilles: How very versantile of him!

Anteater: At the height of his creative powers, he met with a most untimely

demise. One day, a very hot summer day, he was out soaking up the

warmth, when a freak thundershower-the kind that hits only once

every hundred years or so-appeared from out of the blue, and

thoroughly drenched J. S .F. Since the storm came utterly without

warning, the ants got completely disoriented and confused. The intri-

cate organization which had been so finely built up over decades, all

went down the drain in a matter of minutes. It was tragic.

Achilles: Do you mean that all the ants drowned, which obviously would

spell the end of poor J. S. F.?

Anteater: Actually, no. The ants managed to survive, every last one of

them, by crawling onto various sticks and logs which floated above the

raging torrents. But when the waters receded and left the ants back on

their home grounds, there was no organization left. The caste distribu-

tion was utterly destroyed, and the ants themselves had no ability to

reconstruct what had once before been such a finely tuned organiza-

tion. They were as helpless as the pieces of Humpty Dumpty in putting

themselves back together again. I myself tried, like all the king’s horses

and all the king’s men, to put poor Fermant together again. I faithfully

put out sugar and cheese, hoping against hope that somehow Fermant

would reappear ... (Pulls out a handkerchief and wipes his eyes.)

Achilles: How valiant of you! I never knew Anteaters had such big hearts.

Anteater: But it was all to no avail. He was gone, beyond reconstitution.

However, something very strange then began to take place: over the

next few months, the ants which had been components of J. S. F.

slowly regrouped, and built up a new organization. And thus was Aunt

Hillary born.

Crab: Remarkable! Aunt Hillary is composed of the very same ants as

Fermant was?

Anteater: Well, originally she was, yes. By now, some of the older ants have

died, and been replaced. But there are still many holdovers from the

J. S. F.-days.

Crab: And can’t you recognize some of J. S. F.’s old traits coming to the

fore, from time to time, in Aunt Hillary?

Anteater: Not a one. They have nothing in common. And there is no

reason they should, as I see it. There are, after all, often several distinct

ways to rearrange a group of parts to form a "sum". And Aunt

Hillary was just a new "sum" of the old parts. Not MORE than the

sum, mind you-just that particular KIND of sum.

Tortoise: Speaking of sums, I am reminded of number theory, where

occasionally one will be able to take apart a theorem into its component

symbols, rearrange them in a new order, and come up with a new

theorem.

Anteater: I’ve never heard of such a phenomenon, although I confess to

being a total ignoramus in the field.

Achilles: Nor have I heard of it-and I am rather well versed in the field, if

I don’t say so myself. I suspect Mr. T is just setting up one of his

elaborate spoofs. I know him pretty well by now.

Anteater: Speaking of number theory, I am reminded of J. S. F. again, for

number theory is one of the domains in which he excelled. In fact, he

made some rather rema! Kable contributions to number theory. Aunt

Hillary, on the other hand, is remarkably dull-witted in anything that

has even the remotest connection with mathematics. Also, she has only

a rather banal taste in music, whereas Sebastiant was extremely gifted . . In musIC.

Achilles: I am very fond of number theory. Could you possibly relate to us

something of the nature of Sebastiant’s contributions?

Anteater: Very well, then. (Pauses for a moment to sip his tea, then resumes.)

Have you heard of Fourmi’s infamous "Well-Tested Conjecture"?

Achilles: I’m not sure ... It sounds strangely familiar, and yet I can’t quite

place it.

Anteater: It’s a very simple idea. Lierre de Fourmi, a mathematiciant by

vocation but lawyer by avocation, had been reading in his copy of the

classic text Arithmetica by Di of Antus, and came across a page contain-

ing the equation

He immediately realized that this equation has infinitely many solu-

tions a, b, c, and then wrote in the margin the following notorious

comment:

... Ant Fugue

The equation

Na+Nb=Nc

has solutions in positive integers a, b, c, and n only when n = 2 (and

then there are infinitely many triplets a, b, c which satisfy the equa-

tion); but there are no solutions for n > 2. I have discovered a truly

marvelous proof of this statement, which, unfortunately, is so small

that it would be well-nigh invisible if written in the margin.

Ever since that year, some three hundred days ago, mathematiciants

have been vainly trying to do one of two things: either to prove

Fourmi’s claim, and thereby vindicate Fourmi’s reputation, which,

although very high, has been somewhat tarnished by skeptics who

think he never really found the proof he claimed to have found·-or

else to refute the claim, by finding a counterexample: a set of four

integers a, b, c, and n, with n > 2, which satisfy the equation. Until very

recently, every attempt in either direction had met with failure. To be

sure, the Conjecture has been verified for many specific values ofn-in

particular, all n up to 125,000. But no one had succeeded in proving it

for ALL n-no one, that is, until Johant Sebastiant Fermant came upon

the scene. It was he who found the proof that cleared Fourmi’s name.

It now goes under the name "Johant Sebastiant’s Well-Tested Conjec-

ture".

Achilles: Shouldn’t it be called a "Theorem" rather than a "Conjecture", if

it’s finally been given a proper proof?

Anteater: Strictly speaking, you’re right, but tradition has kept it this way.

Tortoise: What sort of music did Sebastiant do?

Anteater: He had great gifts for composition. Unfortunately, his greatest

work is shrouded in mystery, for he never reached the point of pub-

lishing it. Some believe that he had it all in his mind; others are more

unkind, saying that he probably never worked it out at all, but merely

blustered about it.

Achilles: What was the nature of this magnum opus?

Anteater: It was to be a giant prelude and fugue; the fugue was to have

twenty-four voices, and to involve twenty-four distinct subjects, one in

each of the major and minor keys.

Achilles: It would certainly be hard to listen to a twenty-four-voice fugue

as a whole!

Crab: Not to mention composing one!

Anteater: But all that we know of it is Sebastiant’s description of it, which

he wrote in the margin of his copy of Buxtehude’s Preludes and

Fugues for Organ. The last words which he wrote before his tragic

demise were:

I have composed a truly marvelous fugue. In it, I have added

together the power of 24 keys, and the power of 24 themes; I came

up with a fugue with the power of 24 voices. Unfortunately, this

margin is too narrow to contain it.

And the unrealized masterpiece simply goes by the name, "Fermant’s

Last Fugue".

Achilles: Oh, that is unbearably tragic.

Tortoise: Speaking of fugues, this fugue which we have been listening to is

nearly over. Towards the end, there occurs a strange new twist on its

theme. (Flips the page in the Well-Tempered Clavier.) Well, what have

we here? A new illustration-how appealing! (Shows it to the Crab.)

Crab: Well, what have we here? Oh, I see: it’s "HOLISMIONISM", written in

large letters that first shrink and then grow back to their original size.

But that doesn’t make any sens-e, because it’s not a word. Oh me, oh

myl (Passes it to the Anteater.)

Anteater: Well, what have we here? Oh, I see: it’s "REDUCTHOLISM", written

in small letters that first grow and then shrink back to their original

size. But that doesn’t make any sense, because it’s not a word. Oh my,

oh mel (Passes it to Achilles.)

Achilles: I know the rest of you won’t believe this, but in fact this picture

consists of the word "HOLISM" written twice, with the letters continually

shrinking as they proceed from left to right. (Returns it to the Tortoise.)

Tortoise: I know the rest of you won’t believe this, but in fact this picture

consists of the word "REDUCTIONISM" written once, with the letters

continually growing as they proceed from left to right.

Achilles: At last-l heard the new twist on the theme this timel I am so

glad that you pointed it out to me, Mr. Tortoise. Finally, I think I am

beginning to grasp the art of listening to fugues.

GODEL,

ESCHER,

BACH:

an Eternal Golden Braid

Douglas R. Hofstadter

Ant Fugue page 311

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~~~Shapeshifter32~~~