SEHR, volume 4, issue 2: Constructions of the Mind

Updated July 22, 1995

on seeing A's and seeing As

Douglas R. Hofstadter

Because it began life essentially as a branch of the theory of

computation, and because the latter began life essentially as a branch of
logic, the discipline of artificial intelligence (AI) has very deep
historical roots in logic. The English logician George Boole, in the
1850s, was among the first to formulate the idea--in his famous book
The Laws of Thought--that thinking itself follows clear patterns, even
laws, and that these laws could be mathematized. For this reason, I like
to refer to this law-bound vision of the activities of the human mind as
the "Boolean Dream."1
Put more concretely, the Boolean Dream amounts to seeing thinking as
the manipulation of propositions, under the constraint that the rules
should always lead from true statements to other true statements. Note
that this vision of thought places full sentences at center stage. A tacit
assumption is thus that the components of sentences--individual words,
or the concepts lying beneath them--are not deeply problematical
aspects of intelligence, but rather that the mystery of thought is how
these small, elemental, "trivial" items work together in large, complex
(and perforce nontrivial) structures.
To make this more concrete, let me take a few examples from
mathematics, a domain that AI researchers typically focused on in the
early days. A concept like "5" or "prime number" or "definite integral"
would be thought of as trivial or quasi-trivial, in the sense that they are
mere definitions. They would be seen as posing no challenge to a
computer model of mathematical thinking--the cognitive activity of
doing mathematical research. By contrast, dealing with propositions
such as "Every even number greater than 2 is the sum of two prime
numbers," establishing the truth or falsity of which requires work--
indeed, an unpredictable amount of work--would be seen as a deep
challenge. Determining the truth or falsity of such propositions, by
means of formal proof in the framework of an axiomatic system, would
be the task facing a mathematical intelligence. Of course a successful
proof, consisting of many lines, perhaps many pages, of text would be
seen as a very complex cognitive structure, the fruit of an intelligent
machine or mind.
Another domain that appealed greatly to many of the early movers of
AI was chess. Once again, the primitive concepts of chess, such as
"bishop," "diagonal move," "fork," "castling," and so forth were all
seen as similar to mathematical definitions--essential to the game, of
course, but posing little or no mental challenge. In chess, what was felt
to matter was the development of grand strategies involving arbitrarily
complex combinations of these definitional notions. Thus developing
long and intricate series of moves, or playing entire games, was seen as
the important goal.
As might be expected, many of the early AI researchers also enjoyed
mathematical or logical puzzles that involved searching through clearly
defined spaces for subtle sequences or combinations of actions, such as
coin-weighing problems (given a balance, find the one fake coin
among a set of twelve in just three weighings), the missionaries-and-
cannibals puzzle (get three missionaries and three cannibals across a
river in the minimum number of boat trips, under the constraint that
there are never more cannibals than missionaries either on the boat,
which can carry only three people, or on either side of the river),
cryptarithmetic puzzles (find an arithmetically valid replacement for
each letter by some digit in the equation "SEND+MORE=MONEY"),
the Fifteen puzzle (return the fifteen sliding blocks in a four-by-four
array having one movable hole to their original order), or even Rubik's
Cube. All of these involve manipulation of hard-edged components,
and the goal is to find complex sequences of actions that have certain
hard-edged properties. By "hard-edged," I mean that there is no
ambiguity about anything in such puzzles. There is no question about
whether an individual is or is not a cannibal; there is no doubt about the
location of a sliding block; and so forth. Nothing is blurry or vague.
These kinds of early preconceptions about the nature of the challenge
of modeling intelligence on a machine gave a certain clear momentum
to the entire discipline of AI--indeed, deeply influenced the course of
research done all over the world for decades. Nowadays, however, the
tide is slowly turning. Although some work in this logic-rooted
tradition continues to be done, many if not most AI researchers have
reached the conclusion--perhaps reluctantly--that the logic-based formal
approach is a dead end.
What seems to be wrong with it? In a word, logic is brittle, in diametric
opposition with the human mind, which is best described as "flexible"
or "fluid" in its capabilities of dealing with completely new and
unanticipated types of situations. The real world, unlike chess and
some aspects of mathematics, is not hard-edged but ineradicably blurry.
Logic and its many offshoots rely on humans to translate situations into
some unambiguous formal notation before any processing by a
machine can be done. Logic is not at all concerned with such activities
as categorization or the recognition of patterns. And to many people's
surprise, these activities have turned out to play a central role in
intelligence.
It happens that as AI was growing up, a somewhat distinct discipline
called "pattern recognition" (PR) was also being developed, mostly by
different researchers. There was some but not much communication
between the two disciplines. Researchers in PR were concerned with
getting machines to do such things as read handwriting or typewritten
text, visually recognize objects in photographs, and understand spoken
language. In the attempts to get machines to do such things, the
complexity of categories, in its full glory and in its full messiness,
began slowly to emerge. Researchers were faced with questions like
these: What is the essence of dog-ness or house-ness? What is the
essence of 'A'-ness? What is the essence of a given person's face, that it
will not be confused with other people's faces? What is in common
among all the different ways that all different people, including native
speakers and people with accents, pronounce "Hello"? How to convey
these things to computers, which seem to be best at dealing with hard-
edged categories--categories having crystal-clear, perfectly sharp
boundaries?
These kinds of perceptual challenges, despite their formidable, bristling
difficulties, were at one time viewed by most members of the AI
community as a low-level obstacle to be overcome en route to
intelligence--almost as a nuisance that they would have liked to, but
couldn't quite, ignore. For example, the attitude of AI researchers
would be, "Yes, it's damn hard to get a computer to perceive an actual,
three-dimensional chessboard, with all of its roundish shapes, varying
densities of shadows, and so forth, but what does that have to do with
intelligence? Nothing! Intelligence is about finding brilliant chess
moves, something that is done after the perceptual act is completely
over and out of the way. It's a purely abstract thing. Conceptually,
perception and reasoning are totally separable, and intelligence is only
about the latter." In a similar way, the typical AI attitude about doing
math would be that math skill is a completely perception-free activity
without the slightest trace of blurriness--a pristine activity involving
precise, rigid manipulations of the most crystalline of definitions,
axioms, rules of inference--a mental activity that (supposedly) is totally
isolated from, and totally unsullied by, "mere" perception.
These two trends--AI and PR--had almost no overlap. Each group
pursued its own ends with almost no effect on the other group. Very
occasionally, however, one could spot hints of another possible
attitude, radically different from these two. The book Pattern
Recognition, written in the late 1960s by Mikhail Bongard, a Russian
researcher, seemed largely to be a prototypical treatise on pattern
recognition, concerned mostly with recognition of objects and having
little to do with higher mental functioning.2 But then in a splendid
appendix, Bongard revealed his true colors by posing an escalating
series of 100 pattern-recognition puzzles for humans and machines
alike. Each puzzle involved twelve simple line drawings separated into
two sets of six each, and the idea was to figure out what was the basis
for the segregation. What was the criterion for separating the twelve
into these two sets? Readers are invited to try the following Bongard
problem, for instance.

Of course, for each puzzle there were, in a certain trivial sense, an

infinite number of possible solutions. For instance, one could take the
six pictures on the left of any given Bongard problem and say,
"Category 1 contains exactly these six pictures (and no others) and
Category 2 contains all other pictures." This would of course work in a
very literal-minded, heavy-handed way, but it would not be how any
human would ever think of it, except under the most artificial of
circumstances. A psychologically realistic basis for segregation in a
Bongard problem might be that all pictures in Category 1 would
involve no curved lines, say, whereas all pictures in Category 2 would
have at least one curved line. Or another typical segregation criterion
would be that pictures in Category 1 would involve nesting (i.e., the
presence of a shape containing another shape), and pictures in
Category 2 would not. And so on. The following Bongard problems
give a feeling for the kinds of issues that Bongard was concerned with
in his work. Readers are challenged to try to find, for each of them, a
very simple and appealing criterion that distinguishes Category 1 from
Category 2.
The key feature of Bongard problems is that they involve highly
abstract conceptual properties, in strong contrast to the usual tacit
assumption that the quintessence of visual perception is the activity of
dividing a complex scene into its separate constituent objects followed
by the activity of attaching standard labels to the now-separated
objects (i.e., the identification of the component objects as members of
various pre-established categories, such as "car," "dog," "house,"
"hammer," "airplane," etc.). In Bongard problems, by contrast, the
quintessential activity is the discovery of some abstract connection that
links all the various diagrams in one group of six, and that distinguishes
them from all the diagrams in the other group of six. To do this, one
has to bounce back and forth among diagrams, sometimes remaining
within a single set of six, other times comparing diagrams across sets.
But the essence of the activity is a complex interweaving of acts of
abstraction and comparison, all of which involve guesswork rather
than certainty.

By "guesswork," what I mean is that one has to take a chance that

certain aspects of a given diagram matter, and that others are irrelevant.
Perhaps shapes count, but not colors--or vice versa. Perhaps
orientations count, but not sizes--or vice versa. Perhaps curvature or its
lack counts, but not location inside the box--or vice versa. Perhaps
numbers of objects but not their types matter--or vice versa. Somehow,
people usually have a very good intuitive sense, given a Bongard
problem, for which types of features will wind up mattering and which
are mere distractors. Even when one's first hunch turns out wrong, it
often takes but a minor "tweak" of it in order to find the proper aspects
on which to focus. In other words, there is a subtle sense in which
people are often "close to right" even when they are wrong. All of
these kinds of high-level mental activities are what "seeing" the various
diagrams in a Bongard problem--a pattern-recognition activity--
involves.

When presented this way, visual perception takes on a very different

light. Its core seems to be analogy-making--that is, the activity of
abstracting out important features of complex situations (thus filtering
out what one takes to be superficial aspects) and finding resemblances
and differences between situations at that high level of description.
Thus the "annoying obstacle" that AI researchers often took perception
to be becomes, in this light, a highly abstract act--one might even say a
highly abstract art--in which intuitive guesswork and subtle judgments
play the starring roles.

It is clear that in the solution of Bongard problems, perception is

pervaded by intelligence, and intelligence by perception; they
intermingle in such a profound way that one could not hope to tease
them apart. In fact, this phenomenon had already been recognized by
some psychologists, and even celebrated in a rather catchy little slogan:
"Cognition equals perception."

Sadly, Bongard's insights did not have much effect on either the AI
world or the PR world, even though in some sense his puzzles provide
a bridge between the two worlds, and suggest a deep interconnection.
However, they certainly had a far-reaching effect on me, in that they
pointed out that perception is far more than the recognition of members
of already-established categories--it involves the spontaneous
manufacture of new categories at arbitrary levels of abstraction. As I
said earlier, this idea suggested in my mind a profound relationship
between perception and analogy-making--indeed, it suggested that
analogy-making is simply an abstract form of perception, and that the
modeling of analogy-making on a computer ought to be based on
models of perception.

A key event in my personal evolution as an AI researcher was a visit I

made to Carnegie-Mellon University's Computer Science Department
in 1976. While there, I had the good fortune to talk with some of the
developers of the Hearsay II program, whose purpose was to be able to
recognize spoken utterances. They had made an elegant movie to
explain their work, which they showed me. The movie began by
graphically conveying the immense difficulty of the task, and then in
clear pictorial terms showed their strategy for dealing with the problem.

The basic idea was to take a raw speech signal--a waveform, in other
words, which could be seen on a screen as a constantly changing
oscilloscope trace--and to produce from it a hierarchy of "translations"
on different levels of abstraction. The first level above the raw
waveform would thus be a segmented waveform, consisting of an
attempt to break the waveform up into a series of nonoverlapping
segments, each of which would hopefully correspond to a single
phoneme in the utterance. The next level above that would be a set of
phonetic labels attached to each segment, which would serve as a
bridge to the next level up, namely a phonemic hypothesis as to what
phoneme had actually been uttered, such as "o" or "u" or "d" or "t."
Above the phonemic level was the syllabic level, consisting, of course,
in hypothesized syllables such as "min" or "pit" or "blag." Then there
was the word level, which needs little explanation, and above that the
phrase level (containing such hypothesized utterance-fragments as
"when she went there" or "under the table"). One level higher was the
sentence level, which was just below the uppermost level, which was
called the pragmatic level.

At that level, the meaning of the hypothesized sentence was compared

to the situation under discussion (Hearsay always interpreted what it
heard in relation to a specific real-world context such as an ongoing
chess game, not in a vacuum); if it made sense in the given context, it
was accepted, whereas if it made no sense in the context, then some
piece of the hypothesized sentence--its weakest piece, in fact, in a sense
that I will describe below--was modified in such a way as to make the
sentence fit the situation (assuming that such a simple fix was possible,
of course). For example, if the program's best guess as to what it had
heard was the sentence "There's a pen on the box" but in fact, in the
situation under discussion there was a pen that was in a box rather than
on it, and if furthermore the word "on" was the least certain word in the
hypothesized sentence, then a switch to "There's a pen in the box"
might have a high probability of being suggested. If, on the other hand,
the word "on" was very clear and strong whereas the word "pen" was
the least certain element in the sentence, then the sentence might be
converted into "There's a pin on the box." Of course, that sentence
would be suggested as an improvement over the original one only if it
made sense within the context.

This idea of making changes according to expectations (i.e., long-term

knowledge of how the world usually is, as well as the specifics of the
current situation) was a very beautiful one, in my opinion, but it caused
no end of complexity in the program's architecture. In particular, as
soon as the program made a guess at a new sentence--such as
converting "There's a pen on the box" into "There's a pen in the box"--
it took the new word and tried to modify its underpinnings, such as its
syllables, the phonemes below them, their phonetic labels, and possibly
even the boundary lines of segments in the waveform, in an attempt to
see if the revised sentence was in any way justifiable in terms of the
sounds actually produced. If not, it would be rejected, no matter how
strong was its appeal at the pragmatic level. And while all this work
was going on, the program would simultaneously be working on new
incoming waveforms and on other types of possible rehearings of the
old sentence.

The preceding discussion implies that each aspect of the utterance at

each level of abstraction was represented as a type of hypothesis,
attached to which was a set of pieces of evidence supporting the given
hypothesis. Thus attached to a proposed syllable such as "tik" were
little structures indicating the degree of certainty of its component
phonemes, and the probability of correctness of any words in which it
figured. The fact that plausibility values or levels of confidence were
attached to every hypothesis imbued the current best guess with an
implicit "halo" of alternate interpretations, any one of which could step
in if the best guess was found to be inappropriate.

I am sure that the figurative language I am using to describe Hearsay II

would not have been that chosen by its developers, but I am trying to
get across an image that it undeniably created in me, since that image
then formed the nucleus of my own subsequent research projects in AI.
Some other crucial features of the Hearsay II architecture that I have
hinted at but cannot describe here in detail were its deep parallelism, in
which processes of all sorts operated on many levels of abstraction at
the same time, and its uniquely flexible manner of allowing a constant
intermingling of bottom-up processing (i.e., the building-up of higher
levels of abstraction on top of fairly solid lower-level hypotheses, much
like the construction of a building) and top-down processing (i.e., the
attempt to build plausible hypotheses close to the raw data in order to
give a solid underpinning to hypotheses that make sense at abstract
levels, something like constructing lower and lower floors after the top
floors have been built and are sitting suspended in thin air).

Not too surprisingly, my first attempt to turn my personal vision of how

Hearsay II operated into an AI project of my own was the sketching-
out, in very broad strokes, of a hypothetical program to solve Bongard
problems.3 However, the difficulties in actually implementing such a
program completely on my own (this was before I had graduate
students!) seemed so daunting that I backed away from doing so, and
started exploring other domains that seemed more tractable. What I was
always after was some kind of microdomain in which analogies at very
high levels of abstraction could be made, yet which did not require an
extreme amount of real-world knowledge.

Over the years, I developed a number of different computer projects,

each one centered on a different microdomain, and thanks to the hard
work of several superb graduate students, many of these abstract ideas
were converted into genuine working computer programs. All of these
projects are described in considerable detail in the book Fluid Concepts
and Creative Analogies,4 co-authored by me and several of my
students.

Here I would like to present in very quick terms one of those domains
and the challenges that it involved, a project that clearly reveals how
deeply Mikhail Bongard's ideas inspired me. The project's name is
"Letter Spirit," and it is concerned with the visual forms of the letters of
the roman alphabet. In particular, our goal is to build a computer
program that can design all 26 lowercase letters, "a" through "z," in
any number of artistically consistent styles. The task is made even more
"micro" by restricting the letterforms to a grid. In particular, one is
allowed to turn on any of the 56 short horizontal, vertical, and diagonal
line segments--"quanta," as we call them--in the 2´6 array shown
below. By so doing, one can render each of the 26 letters in some
fashion; the idea is to make them all agree with each other stylistically.

To me, it is highly significant that Bongard chose to conclude his

appendix of 100 pattern-recognition problems with a puzzle whose
Category 1 consists of six highly diverse Cyrillic "A"s, and whose
Category 2 consists of six equally diverse Cyrillic "B"s.
This choice of final problem is a symbolic message carrying the clear
implication that, in Bongard's opinion, the recognition of letters
constitutes a far deeper problem than any of his 99 earlier problems--
and the more general conclusion that a necessary prerequisite to
tackling real-world pattern recognition in its infinite complexity is the
development of all the intricate and subtle analogy-making machinery
required to solve his 100 problems and the myriad other ones that lie in
their immediate "halo."

To show the fearsome complexity of the task of letter recognition, I

offer the following display of uppercase "A"s, all designed by
professional typeface designers and used in advertising and similar
functions.
What kind of abstraction could lie behind this crazy diversity? (Indeed,
I once even proposed that the toughest challenge facing AI workers is
to answer the question: "What are the letters 'A' and 'I'?")

The Letter Spirit project attempts to study the conceptual enigma posed
by the foregoing collection, but to do so within the framework of the
grid shown above, and even to extend that enigma in certain ways.
Thus, a Letter Spirit counterpart to the previous illustration would be
the collection of grid-bound lowercase "a"s shown below, suggesting
how intangible the essence of "a"-ness must be, even when the shapes
are made solely by turning on or off very simple, completely fixed line
segments.

I said above that the Letter Spirit project aims not just to study the
enigma of the many "A"s, but to extend that enigma. By this I meant
the following. The challenge of Letter Spirit is not merely the
recognition or classification of a set of given letters, but the creation of
new letterforms, and thereby the creation of new artistic styles. Thus
the task for the program would be to take a given letter designed by a
person--any one of the "a"s below, for instance--and to let that letter
inspire the remaining 25 letters of the alphabet. Thus one might move
down the line consecutively from "a" to "b" to "c," and so on. Of
course, the seed letter need not be an "a," and even if it were an "a,"
the program would be very unlikely to proceed in strict alphabetical
order (if one has created an "h," it is clearly more natural to try to
design the "n" before tackling the design of "i"); but let us nonetheless
imagine a strictly alphabetic design process stopped while under way,
so that precisely the first seven letters of the alphabet have been
designed, and the remaining nineteen remain to be done. Let us in fact
imagine doing such a thing with seven quite different initial "a"s. We
would thus have something like the 7´7 matrix shown below.
Implicit in this matrix (especially in the dot-dot-dots on the right side
and at the bottom) are two very deep pattern-recognition problems.
First is the "vertical problem"--namely, what do all the items in any
given column have in common? This is essentially the question that
Bongard was asking in the final puzzle of his appendix. The answer, in
a single word, is: Letter. Of course, to say that one word is not to solve
the problem, but it is a useful summary. The second problem is, of
course, the "horizontal problem"--namely, what do all the items in any
given row have in common? To this question, I prefer the single-word
answer: Spirit. How can a human or a machine make the uniform
artistic spirit lurking behind these seven shapes leap to the abstract
category of "h," then leap from those eight shapes to the category "i,"
then leap to "j," and so on, all the way down the line to "z"?

And do not think that "z" is really the end of the line. After all, there
remain all the uppercase letters, and then all the numerals, and then
punctuation marks, and then mathematical symbols... But even this is
not the end, for one can try to make the same spirit leap out of the
roman alphabet and into such other writing systems as the Greek
alphabet, the Russian alphabet, Hebrew, Japanese, Arabic, Chinese,
and on and on. Of course, the making of such "transalphabetic
leaps" (as I like to call them) goes way beyond the modest limits of the
Letter Spirit project itself, but the suggestion serves as a reminder that,
just as there are unimaginably many different spirits (i.e., artistic styles)
in which to realize any given letter of the alphabet, there are also
unimaginably many different "letters" (i.e., typographical categories) in
which to realize any given stylistic spirit.

In metaphorical terms, one can talk about the alphabet and the
"stylabet"--the set of all conceivable styles. Both of these "bets" are
infinite rather than finite entities. The stylabet is very much like the
alphabet in its subtlety and intangibility, but it resides at a considerably
higher level of abstraction.

The one-word answers to the so-called vertical and horizontal

questions--"letter" and "spirit"--gave rise to the project's name. There is
of course a classic opposition in the legal domain between the concepts
of "letter" and "spirit"--the contrast between "the letter of the law" and
"the spirit of the law." The former is concrete and literal, the latter
abstract and spiritual. And yet there is a continuum between them. A
given law can be interpreted at many levels of abstraction. So too with
the artistic design problems of the Letter Spirit project: there are many
ways to extrapolate from a given seed letter to other alphabetic
categories, some ways being rather simplistic and down-to-earth, others
extremely sophisticated and high-flown. The Letter Spirit project does
not by any means grow out of the dubious postulate that there is one
unique "best" way to carry style consistently from one category to
another; rather, it allows many possible notions of artistically valid style
at many different levels of abstraction. Of course this means that the
project is in complete opposition to any view of intelligence that sees
the main purpose of mind as being an eternal quest after "right
answers" and "truth." That the human mind can conduct such a quest,
principally through such careful disciplines as mathematics, science,
history, and so forth, is a tribute to its magnificent subtlety, but to do
science and history is not how or why the mind evolved, and it deeply
misrepresents the mind to cast its activities solely in the narrow and
rigid terms of truth-seeking.

To convey something of the flavor of the Letter Spirit project, I offer

the following sample style-extrapolation puzzle, which I hope will
intrigue readers. Take the following gridbound way of realizing the
letter "d" and attempt to make a letter "b" that exhibits the same spirit,
or style.

One idea that springs instantly to mind for many people is simply to
reflect the given shape, since one tends to think of "d" and "b" as being
in some sense each other's mirror images. For many "d"s, this simple
recipe for making a "b" might work, but in this case there is a
somewhat troubling aspect to the proposal: the resultant shape has quite
an "h"-ish look to it, enough perhaps to give a careful letter designer
second thoughts.
What escape routes might be found, still respecting the rigid constraints
of the grid?

One possible idea is that of reversing the direction of the two diagonal
quanta at the bottom, to see if that action reduces the "h"-ishness.

To some people's eyes, including mine, this action slightly improves

the ratio of "b"-ness to "h"-ness. Notice that this move also has the
appealing feature of echoing the exact diagonals of the seed letter. This
agreement could be taken as a particular type of stylistic consistency.
Perhaps, then, this is a good enough "b," but perhaps not.

Another way one might try to entirely sidestep "h"-ishness would

involve somehow shifting the opening from the bottom to the top of the
bowl. Can you find a way to carry this out? Or are there yet other
possibilities?

I must emphasize that this is not a puzzle with a clearly optimal answer;
it is posed simply as an artistic challenge, to try to get across the nature
of the Letter Spirit project. When you have made a "b" that satisfies
you, can you proceed to other letters of the alphabet? Can you make an
entire alphabet? How does your set of 26 letters, all inspired by the
given seed letter, compare with someone else's?
The Letter Spirit project is doubtless the most ambitious project in the
modeling of analogy-making and creativity so far undertaken in my
research group, and as of this writing, it has by no means been fully
realized as a computer program. It is currently somewhere between a
sketch and a working program, and in perhaps a couple of years a
preliminary version will exist. But it builds upon several already-
realized programs, all of whose architectures were deeply inspired by
the ideas of Mikhail Bongard and by principles derived from the
architecture of the pioneering perceptual program Hearsay II.

To conclude, I would like to cite the words of someone whose fluid

way of thinking I have always admired--the great mathematician
Stanislaw Ulam. As Heinz Pagels reports in his book The Dreams of
Reason, one time Ulam and his mathematician friend Gian-Carlo Rota
were having a lively debate about artificial intelligence, a discipline
whose approach Ulam thought was simplistic. Convinced that
perception is the key to intelligence, Ulam was trying to explain the
subtlety of human perception by showing how subjective it is, how
influenced by context. He said to Rota, "When you perceive
intelligently, you always perceive a function, never an object in the
physical sense. Cameras always register objects, but human perception
is always the perception of functional roles. The two processes could
not be more different.... Your friends in AI are now beginning to
trumpet the role of contexts, but they are not practicing their lesson.
They still want to build machines that see by imitating cameras,
perhaps with some feedback thrown in. Such an approach is bound to
fail..."

Rota, clearly much more sympathetic than Ulam to the old-fashioned

view of AI, interjected, "But if what you say is right, what becomes of
objectivity, an idea formalized by mathematical logic and the theory of
sets?"

Ulam parried, "What makes you so sure that mathematical logic

corresponds to the way we think? Logic formalizes only a very few of
the processes by which we actually think. The time has come to enrich
formal logic by adding to it some other fundamental notions. What is it
that you see when you see? You see an object as a key, a man in a car
as a passenger, some sheets of paper as a book. It is the word 'as' that
must be mathematically formalized.... Until you do that, you will not
get very far with your AI problem."

To Rota's expression of fear that the challenge of formalizing the

process of seeing a given thing as another thing was impossibly
difficult, Ulam said, "Do not lose your faith--a mighty fortress is our
mathematics," a droll but ingenious reply in which Ulam practices what
he is preaching by seeing mathematics itself as a fortress!

If anyone else but Stanislaw Ulam had made the claim that the key to
understanding intelligence is the mathematical formalization of the
ability to "see as," I would have objected strenuously. But knowing
how broad and fluid Ulam's conception of mathematics was, I think he
would have been able to see the Letter Spirit architecture and its
predecessor projects as mathematical formalizations.

In any case, when I look at Ulam's key word "as," I see it as an

acronym for "Abstract Seeing" or perhaps "Analogical Seeing." In this
light, Ulam's suggestion can be restated in the form of a dictum--"Strive
always to see all of AI as AS"--a rather pithy and provocative slogan to
which I fully subscribe.

Notes

1 For more on this, see "Waking Up from the Boolean Dream,"

Chapter 26 of my book, Metamagical Themas (New York: Basic,
1985).

2 See Mikhail Moiseevich Bongard, Pattern Recognition (New York:

Spartan Books, 1970).

3 See Chapter 19 of my book Gödel, Escher, Bach (New York: Basic,

1979) for this sketched architecture.

4 Douglas R. Hofstadter and the Fluid Analogies Research Group,

Fluid Concepts and Creative Analogies: Computer Models of the
Fundamental Mechanism of Thought (New York: Basic, 1995).