Below is the unedited preprint (not a quotable final draft) of:
Boden, Margaret A. (1994). Precis of The creative mind: Myths and mechanisms. Behavioral and Brain Sciences 17 (3): 519-570.
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Precis of "THE CREATIVE MIND: MYTHS AND MECHANISMS" London: Weidenfeld & Nicolson 1990 (Expanded edn., London: Abacus, 1991.)

Margaret A. Boden
School of Cognitive and Computing Sciences
University of Sussex
England FAX: 0273-671320
maggieb@syma.susx.ac.uk

Keywords

creativity, intuition, discovery, association, induction, representation, unpredictability, artificial intelligence, computer music, story-writing, computer art, Turing test

Abstract

What is creativity? One new idea may be creative, while another is merely new: what's the difference? And how is creativity possible? -- These questions about human creativity can be answered, at least in outline, using computational concepts.

There are two broad types of creativity, improbabilist and impossibilist. Improbabilist creativity involves (positively valued) novel combinations of familiar ideas. A deeper type involves METCS: the mapping, exploration, and transformation of conceptual spaces. It is impossibilist, in that ideas may be generated which -- with respect to the particular conceptual space concerned -- could not have been generated before. (They are made possible by some transformation of the space.) The more clearly conceptual spaces can be defined, the better we can identify creative ideas. Defining conceptual spaces is done by musicologists, literary critics, and historians of art and science. Humanist studies, rich in intuitive subtleties, can be complemented by the comparative rigour of a computational approach.

Computational modelling can help to define a space, and to show how it may be mapped, explored, and transformed. Impossibilist creativity can be thought of in "classical" AI-terms, whereas connectionism illuminates improbabilist creativity. Most AI-models of creativity can only explore spaces, not transform them, because they have no self-reflexive maps enabling them to change their own rules. A few, however, can do so.

A scientific understanding of creativity does not destroy our wonder at it, nor make creative ideas predictable. Demystification does not imply dehumanization.


Chapter 1: The Mystery of Creativity

Creativity surrounds us on all sides: from composers to chemists, cartoonists to choreographers. But creativity is a puzzle, a paradox, some say a mystery. Inventors, scientists, and artists rarely know how their original ideas arise. They mention intuition, but cannot say how it works. Most psychologists cannot tell us much about it, either. What's more, many people assume that there will never be a scientific theory of creativity -- for how could science possibly explain fundamental novelties? As if all this were not daunting enough, the apparent unpredictability of creativity seems (to many people) to outlaw any systematic explanation, whether scientific or historical.

Why does creativity seem so mysterious? Artists and scientists typically have their creative ideas unexpectedly, with little if any conscious awareness of how they arose. But the same applies to much of our vision, language, and common-sense reasoning. Psychology includes many theories about unconscious processes. Creativity is mysterious for another reason: the very concept is seemingly paradoxical.

If we take seriously the dictionary-definition of creation, "to bring into being or form out of nothing", creativity seems to be not only beyond any scientific understanding, but even impossible. It is hardly surprising, then, that some people have "explained" it in terms of divine inspiration, and many others in terms of some romantic intuition, or insight. From the psychologist's point of view, however, "intuition" is the name not of an answer, but of a question. How does intuition work?

In this book, I argue that these matters can be better understood, and some of these questions answered, with the help of computational concepts.

This claim in itself may strike some readers as absurd, since computers are usually assumed to have nothing to do with creativity. Ada Lovelace is often quoted in this regard: "The Analytical Engine has no pretensions whatever to originate anything. It can do [only] whatever we know how to order it to perform." If this is taken to mean that a computer can do only what its program enables it to do, it is of course correct. But it does not follow that there can be no interesting relations between creativity and computers.

We must distinguish four different questions, which are often confused with each other. I call them Lovelace questions, and state them as follows:

(1) Can computational concepts help us to understand human creativity?

(2) Could a computer, now or in the future, appear to be creative?

(3) Could a computer, now or in the future, appear to recognize creativity?

(4) Could a computer, however impressive its performance, really be creative?

The first three of these are empirical, scientific, questions. In Chapters 3-10, I argue that the answer to each of them is "Yes". (The first Lovelace question is discussed in each of those chapters; in chapters 7-8, the second and third are considered also.)

The fourth Lovelace question is not a scientific enquiry, but a philosophical one. (More accurately, it is a mix of three complex, and highly controversial, philosophical problems.) I discuss it in Chapter 11. However, one may answer "Yes" to the first three Lovelace questions without necessarily doing so for the fourth. Consequently, the fourth Lovelace question is ignored in the main body of the book, which is concerned rather with the first three Lovelace questions.

Chapter 2: The Story so Far

This chapter draws on some of the previous literature on creativity. But it is not a survey. Its aim is to introduce the main psychological questions, and some of the historical examples, addressed in detail later in the book. The main writers mentioned are Poincare (1982), Hadamard (1954) Koestler (1975), and Perkins (1981).

Among the points of interest in Poincare's work are his views on associative memory. He described our ideas as "something like the hooked atoms of Epicurus," flashing in every direction like "a swarm of gnats, or the molecules of gas in the kinematic theory of gases". He was well aware that how the relevant ideas are aroused, and how they are joined together, are questions which he could not answer in detail. Another interesting aspect of Poincare's approach is his distinction between four "phases" of creativity, some conscious some unconscious.

These four phases were later named (by Hadamard) as preparation, incubation, inspiration and verification (evaluation). Hadamard, besides taking up Poincare's fourfold distinction, spoke of finding problem-solutions "quite different" from any he had previously tried. If (as Poincare had claimed) the gnat-like ideas were only "those from which we might reasonably expect the desired solution", then how could such a thing happen?

Perkins has studied the four phases, and criticizes some of the assumptions made by Poincare and Hadamard. In addition, he criticizes the romantic notion that creativity is due to some special gift. Instead, he argues that "insight" involves everyday psychological capacities, such as noticing and remembering. (The "everyday" nature of creativity is discussed in Chapter 10.)

Koestler's view that creativity involves "the bisociation of matrices" comes closest to my own approach. However, his notion is very vague. The body of my book is devoted to giving a more precise account of the structure of "matrices" (of various kinds), and of just how they can be "bisociated" so as to result in a novel idea -- sometimes (as in Hadamard's experience) one quite different from previous ideas. (Matrices appear in my terminology as conceptual spaces, and different forms of bisociation as association, analogy, exploration, or transformation.)

Among the examples introduced here are Kekule's discovery of the cyclical structure of the benzene molecule, Kepler's (and Copernicus') thoughts on elliptical orbits, and Coleridge's poetic imagery in Kubla Khan. Others mentioned in passing include Coleridge's announced intention to write a poem about an ancient mariner, Bach's harmonically systematic set of preludes and fugues, the jazz-musician's skill in improvising a melody to fit a chord sequence, and our everyday ability to recognize that two different apples fall into the same class. All these examples, and many others, are mentioned in later chapters.

Chapter 3: Thinking the Impossible

Given the seeming paradoxicality of the concept of creativity (noted in Chapter 1), we need to define it carefully before going further. This is not straightforward (over 60 definitions appear in the psychological literature (Taylor, 1988)). Part of the reason for this is that creativity is not a natural kind, such that a single scientific theory could explain every case. We need to distinguish "improbabilist" and "impossibilist" creativity, and also "psychological" and "historical" creativity.

People of a scientific cast of mind, anxious to avoid romanticism and obscurantism, generally define creativity in terms of novel combinations of familiar ideas. Accordingly, the surprise caused by a creative idea is said to be due to the improbability of the combination. Many psychometric tests designed to measure creativity work on this principle.

The novel combinations must be valuable in some way, because to call an idea creative is to say that it is not only new, but interesting. However, combination-theorists often omit value from their definition of creativity (although psychometricians may make implicit value-judgements when scoring the novel combinations produced by their experimental subjects). A psychological explanation of creativity focusses primarily on how creative ideas are generated, and only secondarily on how they are recognized as being valuable. As for what counts as valuable, and why, these are not purely psychological questions. They also involve history, sociology, and philosophy, because value-judgments are largely culture-relative (Brannigan, 1981; Schaffer, in press.) Even so, positive evaluation should be explicitly mentioned in definitions of creativity.

Combination-theorists may think they are not only defining creativity, but explaining it, too. However, they typically fail to explain how it was possible for the novel combination to come about. They take it for granted, for instance, that we can associate similar ideas and recognize more distant analogies, without asking just how such feats are possible. A psychological theory of creativity needs to explain how associative and analogical thinking works (matters discussed in Chapters 6 and 7, respectively).

These two cavils aside, what is wrong with the combination-theory? Many ideas which we regard as creative are indeed based on unusual combinations. For instance, the appeal of Heath-Robinson machines lies in the unexpected uses of everyday objects; and poets often delight us by juxtaposing seemingly unrelated concepts. For creative ideas such as these, a combination-theory, supplemented by psychological explanations of association and analogy, might suffice.

Many creative ideas, however, are surprising in a deeper way. They concern novel ideas that not only did not happen before, but which -- we intuitively feel -- could not have happened before.

Before considering just what this "could not" means, we must distinguish two further senses of creativity. One is psychological, or personal: I call it P-creativity. The other is historical: H-creativity. The distinction between P-creativity and H-creativity is independent of the improbabilist/impossibilist distinction made above: all four combinations occur. However, I use the P/H distinction primarily to compare cases of impossibilist creativity.

Applied to impossibilist examples, a valuable idea is P-creative if the person in whose mind it arises could not (in the relevant sense of "could not") have had it before. It does not matter how many times other people have already had the same idea. By contrast, a valuable idea is H-creative if it is P-creative and no-one else, in all human history, has ever had it before.

H-creativity is something about which we are often mistaken. Historians of science and art are constantly discovering cases where other people have had an idea popularly attributed to some national or international hero. Even assuming that the idea was valued at the time by the individual concerned, and by some relevant social group, our knowledge of it is largely accidental. Whether an idea survives, and whether historians at a given point in time happen to have evidence of it, depend on a wide variety of unrelated factors. These include flood, fashion, rivalries, illness, trade-patterns, and wars.

It follows that there can be no systematic explanation of H-creativity, no theory that explains all and only H-creative ideas. For sure, there can be no psychological explanation of this historical category. But all H-creative ideas, by definition, are P-creative too. So a psychological explanation of P-creativity would include H-creative ideas as well.

What does it mean to say that an idea "could not" have arisen before? Unless we know that, we cannot make sense of P-creativity (or H-creativity either), for we cannot distinguish radical novelties from mere "first-time" newness.

An example of a novelty that clearly could have happened before is a newly-generated sentence, such as "The deckchairs are on the top of the mountain, three miles from the artificial flowers". I have never thought of that sentence before, and probably no-one else has, either. Chomsky remarked on this capacity of language-speakers to generate first-time novelties endlessly, and called language "creative" accordingly. But the word "creative" was ill-chosen. Novel though the sentence about deckchairs is, there is a clear sense in which it could have occurred before. For it can be generated by any competent speaker of English, following the same rules that can generate other English sentences. To come up with a new sentence, in general, is not to do something P-creative.

The "coulds" in the previous paragraph are computational "coulds". In other words, they concern the set of structures (in this case, English sentences) described and/or produced by one and the same set of generative rules (in this case, English grammar). There are many sorts of generative system: English grammar is like a mathematical equation, a rhyming-schema for sonnets, the rules of chess or tonal harmony, or a computer program. Each of these can (timelessly) describe a certain set of possible structures. And each might be used, at one time or another, in actually producing those structures.

Sometimes, we want to know whether a particular structure could, in principle, be described by a specific schema, or set of abstract rules. -- Is "49" a square number? Is 3,591,471 a prime? Is this a sonnet, and is that a sonata? Is that painting in the Impressionist style? Could that geometrical theorem be proved by Euclid's methods? Is that word-string a sentence? Is a benzene-ring a molecular structure describable by early nineteenth-century chemistry (before Kekule had his famous vision in 1865)? -- To ask whether an idea is creative or not (as opposed to how it came about) is to ask this sort of question.

But whenever a structure is produced in practice, we can also ask what generative processes actually went on in its production. -- Did a particular geometer prove a particular theorem in this way, or in that? Was the sonata composed by following a textbook on sonata-form? Did Kekule rely on the then-familiar principles of chemistry to generate his seminal idea of the benzene-ring, and if not how did he come up with it? -- To ask how an idea (creative or otherwise) actually arose, is to ask this type of question.

We can now distinguish first-time novelty from impossibilist originality. A merely novel idea is one which can be described and/or produced by the same set of generative rules as are other, familiar, ideas. A genuinely original, or radically creative, idea is one which cannot. It follows that the ascription of (impossibilist) creativity always involves tacit or explicit reference to some specific generative system.

It follows, too, that constraints -- far from being opposed to creativity -- make creativity possible. To throw away all constraints would be to destroy the capacity for creative thinking. Random processes alone, if they happen to produce anything interesting at all, can result only in first-time curiosities, not radical surprises. (As explained in Chapter 9, randomness can sometimes contribute to creativity -- but only in the context of background constraints.)

Chapter 4: Maps of the Mind

The definition of (impossibilist) creativity given in Chapter 3 implies that, with respect to the usual mental processing in the relevant domain (chemistry, poetry, music ...), a creative idea may be not just improbable, but impossible. How could it arise, then, if not by magic? And how can one impossible idea be more surprising, more creative, than another? If an act of creation is not mere combination, what is it? How can such creativity possibly happen?

To understand this, we need to think of creativity in terms of the mapping, exploration, and transformation of conceptual spaces. (The notion of a conceptual space is used informally in this chapter; later, we see how conceptual spaces can be described more rigorously.) A conceptual space is a style of thinking. Its dimensions are the organizing principles which unify, and give structure to, the relevant domain. In other words, it is the generative system which underlies that domain and which defines a certain range of possibilities: chess-moves, or molecular structures, or jazz-melodies.

The limits, contours, pathways, and structure of a conceptual space can be mapped by mental representations of it. Such mental maps can be used (not necessarily consciously) to explore -- and to change -- the spaces concerned.

Evidence from developmental psychology supports this view. Children's skills are at first utterly inflexible. Later, imaginative flexibility results from "representational redescriptions" (RRs) of (fluent) lower-level skills (Clark & Karmiloff-Smith, in press; Karmiloff-Smith, 1993). These RRs provide many-levelled maps of the mind, which are used by the subject to do things he or she could not do before.

For example, children need RRs of their lower-level drawing-skills in order to draw non-existent, or "funny", objects: a one-armed man, or seven-legged dog. Lacking such cognitive resources, a 4-year-old simply cannot spontaneously draw a one-armed man, and finds it very difficult even to copy a drawing of a two-headed man. But 10-year-olds can explore their own man-drawing skill, by using strategies such as distorting, repeating, omitting, or mixing parts. These imaginative strategies develop in a fixed order: children can change the size or shape of an arm before they can insert an extra one, and long before they can give the drawn man wings in place of arms.

The development of RRs is a mapping-exercise, whereby people develop explicit mental representations of knowledge already possessed implicitly.

Few AI-models of creativity contain reflexive descriptions of their own procedures, and/or ways of varying them. Accordingly, most AI-models are limited to exploring their conceptual spaces, rather than transforming them (see Chapters 7 & 8).

Conceptual spaces can be explored in various ways. Some exploration merely shows us something about the nature of the relevant conceptual space which we had not explicitly noticed before. When Dickens described Scrooge as "a squeezing, wrenching, grasping, scraping, clutching, covetous old sinner", he was exploring the space of English grammar. He was reminding the reader (and himself) that the rules of grammar allow us to use seven adjectives before a noun. That possibility already existed, although its existence may not have been realized by the reader.

Some exploration, by contrast, shows us the limits of the space, and identifies specific points at which changes could be made in one dimension or another. To overcome a limitation in a conceptual space, one must change it in some way. One may also change it, of course, without yet having come up against its limits. A small change (a "tweak") in a relatively superficial dimension of a conceptual space is like opening a door to an unvisited room in an existing house. A large change (a "transformation"), especially in a relatively fundamental dimension, is more like the instantaneous construction of a new house, of a kind fundamentally different from (albeit related to) the first.

A complex example of structural exploration and change can be found in the development of post-Renaissance Western music, based on the generative system known as tonal harmony. From its origins to the end of the nineteenth century, the harmonic dimensions of this space were continually tweaked to open up the possibilities (the rooms) implicit in it from the start. Finally, a major transformation generated the deeply unfamiliar (yet closely related) space of atonality.

Each piece of tonal music has a "home-key", from which it starts, from which (at first) it did not stray, and in which it must finish. Reminders of the home-key were constantly provided, as fragments of scales, chords. or arpeggios. As time passed, the range of possible home-keys became increasingly well-defined (Bach's "Forty-Eight" was designed to explore, and clarify, the tonal range of the well-tempered keys).

Soon, travelling along the path of the home-key alone became insufficiently challenging. Modulations between keys were then allowed, within the body of the composition. At first, only a small number of modulations (perhaps only one, followed by its "cancellation") were tolerated, between strictly limited pairs of harmonically-related keys. Over the years, the modulations became more daring, and more frequent -- until in the late nineteenth century there might be many modulations within a single bar, not one of which would have appeared in early tonal music. The range of harmonic relations implicit in the system of tonality gradually became apparent. Harmonies that would have been unacceptable to the early musicians, who focussed on the most central or obvious dimensions of the conceptual space, became commonplace.

Moreover, the notion of the home-key was undermined. With so many, and so daring, modulations within the piece, a "home-key" could be identified not from the body of the piece, but only from its beginning and end. Inevitably, someone (it happened to be Schoenberg) eventually suggested that the convention of the home-key be dropped altogether, since it no longer constrained the composition as a whole. (Significantly, Schoenberg suggested new musical constraints: using every note in the chromatic scale, for instance.)

However, exploring a conceptual space is one thing: transforming it is another. What is it to transform such a space?

One example has just been mentioned: Schoenberg's dropping the home-key constraint to create the space of atonal music. Dropping a constraint is a general heuristic, or method, for transforming conceptual spaces. The deeper the generative role of the constraint in the system concerned, the greater the transformation of the space. Non-Euclidean geometry, for instance, resulted from dropping Euclid's fifth axiom.

Another very general way of transforming conceptual spaces is to "consider the negative": that is, to negate a constraint. One well-known instance concerns Kekule's discovery of the benzene-ring. He described it like this:

"I turned my chair to the fire and dozed. Again the atoms were gambolling before my eyes.... [My mental eye] could distinguish larger structures, of manifold conformation; long rows, sometimes more closely fitted together; all twining and twisting in snakelike motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke."

This vision was the origin of his hunch that the benzene-molecule might be a ring, a hunch that turned out to be correct. Prior to this experience, Kekule had assumed that all organic molecules are based on strings of carbon atoms. But for benzene, the valencies of the constituent atoms did not fit.

We can understand how it was possible for him to pass from strings to rings, as plausible chemical structures, if we assume three things (for each of which there is independent psychological evidence). First, that snakes and molecules were already associated in his thinking. Second, that the topological distinction between open and closed curves was present in his mind. And third, that the "consider the negative" heuristic was present also. Taken together, these three factors could transform "string" into "ring".

A string-molecule is an open curve: one having at least one end-point (with a neighbour on only one side). If one considers the negative of an open curve, one gets a closed curve. Moreover, a snake biting its tail is a closed curve which one had expected to be an open one. For that reason, it is surprising, even arresting ("But look! What was that?"). Kekule might have had a similar reaction if he had been out on a country walk and happened to see a snake with its tail in its mouth. But there is no reason to think that he would have been stopped in his tracks by seeing a Victorian child's hoop. A hoop is a hoop, is a hoop: no topological surprises there. (No topological surprises in a snaky sine-wave, either: so two intertwined snakes would not have interested Kekule, though they might have stopped Francis Crick dead in his tracks, a century later.)

Finally, the change from open curves to closed ones is a topological change, which by definition will alter neighbour-relations. And Kekule was an expert chemist, who knew very well that the behaviour of a molecule depends partly on how the constituent atoms are juxtaposed. A change in atomic neighbour-relations is very likely to have some chemical significance. So it is understandable that he had a hunch that this tail-biting snake-molecule might contain the answer to his problem.

Plausible though this talk of conceptual spaces may be, it is -- thus far -- largely metaphorical. I have claimed that in calling an idea creative one should specify the particular set of generative principles with respect to which it is impossible. But I have not said how the (largely tacit) knowledge of literary critics, musicologists, and historians of art and science might be explicitly expressed within a psychological theory of creativity. Nor have I said how we can be sure that the mental processes specified by the psychologist really are powerful enough to generate such-and-such ideas from such-and-such structures.

This is where computational psychology can help us. I noted above, for example, that representational redescription develops explicit mental representations of knowledge already possessed implicitly. In computational terms, one could -- and Karmiloff-Smith does -- put this by saying that knowledge embedded in procedures becomes available, after redescription, as part of the system's data-structures. Terms like procedures and data-structures are well understood, and help us to think clearly about the mapping and negotiation of conceptual spaces. In general, whatever computational psychology enables us to say, it enables us to say relatively clearly.

Moreover, computational questions can be supplemented by computational models. A functioning computer program, in effect, enables the system to use its maps not just to contemplate the relevant conceptual territory, but to explore it actively. So as well as saying what a conceptual space is like (by mapping it), we can get some clear ideas about how it is possible to move around within it. In addition, those (currently, few) AI-models of creativity which contain reflexive descriptions of their own procedures, and ways of varying them, can transform their own conceptual spaces, as well as exploring them.

The following chapters, therefore, employ a computational approach in discussing the account of creativity introduced in Chapters 1-4.

Chapter 5: Concepts of Computation

Computational concepts drawn from "classical" (as well as connectionist) AI can help us to think about the nature, form, and negotiation of conceptual spaces. Examples of such concepts, most of which were inspired by pre-existing psychological notions in the first place, include the following: generative system, heuristic (both introduced in previous chapters), effective procedure, search-space, search-tree, knowledge representation, semantic net, scripts, frames, what-ifs, and analogical representation.

Each of these concepts is briefly explained in Chapter 5, for people who (unlike BBS-readers) may know nothing about AI or computational psychology. And they are related to a wide range of everyday and historical examples -- some of which will be mentioned again in later chapters.

My main aim, here, is to encourage the reader to use these concepts in considering specific cases of human thought. A secondary aim is to blur the received distinction between "the two cultures". The differences between creativity in art and science lie less in how new ideas are generated than in how they are evaluated, once they have arisen. The uses of computational concepts in this chapter are informal, even largely metaphorical. But in bringing a computational vocabulary to bear on a variety of examples, the scene is set for more detailed consideration (in Chapters 6-8) of some computer models of creativity.

In Chapter 5, I refer very briefly to a few AI-programs (such as chess-machines and Schankian question-answering programs). Only two are discussed at any length: Longuet-Higgins' (1987) work on the perception of tonal harmony, and Gelernter's (1963) geometry (theorem-proving) machine.

Longuet-Higgins' work is not intended as a model of musical creativity. Rather, it provides (in my terminology) a map of a certain sort of musical space: the system of tonal harmony introduced in Chapter 4. In addition, it suggests some ways of negotiating that space, for it identifies musical heuristics that enable the listener to appreciate the structure of the composition. Just as speech perception is not the same as speech production, so appreciating music is different from composing it. Nevertheless, some of the musical constraints that face composers working in this particular genre have been identified in this work.

I also mention Longuet-Higgins' recent work on musical expressiveness, but do not describe it here. In (Boden, in press), I say a little more about it. Without expression, music sounds "dead", even absurd. In playing the notes in a piano-score, for instance, pianists add such features as legato, staccato, piano, forte, sforzando, crescendo, diminuendo, rallentando, accelerando, ritenuto, and rubato. But how? Can we express this musical sensibility precisely? That is, can we specify the relevant conceptual space?

Longuet-Higgins (in preparation), using a computational method, has tried to specify the musical skills involved in playing expressively. Working with two of Chopin's piano-compositions, he has discovered some counterintuitive facts. For example, a crescendo is not uniform, but exponential (a uniform crescendo does not sound like a crescendo at all, but like someone turning-up the volume-knob on a wireless); similarly, a rallentando must be exponentially graded (in relation to the number of bars in the relevant section) if it is to sound "right". Where sforzandi are concerned, the mind is highly sensitive: as little as a centisecond makes a difference between acceptable and clumsy performance.

This work is not a study of creativity. It does not model the exploration of a conceptual space, never mind its transformation. But it is relevant because creativity can be ascribed to an idea (including a musical performance) only by reference to a particular conceptual space. The more clearly we can map this space, the more confidently we can identify and ask questions about the creativity involved in negotiating it. A pianist whose playing-style sounds "original", or even "idiosyncratic", is exploring and transforming the space of expressive skills which Longuet-Higgins has studied.

Gelernter's program, likewise, was not focussed on creativity as such. (It was not even intended as a model of human psychology.) Rather, it was an early exercise in automatic problem-solving, in the domain of Euclidean geometry. However, it is well known that the program was capable of generating a highly elegant proof (that the base-angles of an isosceles triangle are equal), whose H-creator was the fourth-century mathematician Pappus.

Or rather, it is widely believed that Gelernter's program could do this. The ambiguity, not to say the mistake, arises because the program's proof is indeed the same as Pappus' proof, when both are written down on paper in the style of a geometry text-book. But the (creative) mental processes by which Pappus did this, and by which the modern geometer is able to appreciate the proof, were very different from those in Gelernter's program -- which were not creative at all.

Consider (or draw) an isosceles triangle ABC, with A at the apex. You are required to prove that the base-angles are equal. The usual method of proving this, which the program was expected to employ, is to construct a line bisecting angle BAC, running from A to D (a point on the baseline, BC). Then, the proof goes as follows:

Consider triangles ABD and ACD.

AB = AC (given)

AD = DA (common)

Angle BAD = angle DAC (by construction)

Therefore the two triangles are congruent (two sides and included angle equal)

Therefore angle ABD = angle ACD.

Q.E.D.

By contrast, the Gelernter proof involved no construction, and went as follows:

Consider triangles ABC and ACB.

Angle BAC = angle CAB (common)

AB = AC (given)

AC = AB (given)

Therefore the two triangles are congruent (two sides and included angle equal)

Therefore angle ABC = angle ACB.

Q.E.D.

And, written down on paper, this is the outward form of Pappus' proof, too.

The point, here, is that Pappus' own notes (as well as the reader's geometrical intuitions) show that in order to produce or understand this proof, a human being considers one and the same triangle rotated (as Pappus put it, lifted up and replaced in the trace left behind by itself). There were thus two creative aspects of this proof. First, when "congruence" is in question, the geometer normally thinks of two entirely separate triangles (or, sometimes, two distinct triangles having one side in common). Second, Euclidean geometry deals only with points, lines, and planes -- so one would expect any proof to be restricted to two spatial dimensions. But Pappus (and you, when you thought about this proof) imagined lifting and rotating the triangle in the third dimension. He was, if you like, cheating. However, to transform a rule (an aspect of some conceptual space) is to change it: in effect, to cheat. In that sense, transformational creativity always involves cheating.

Gelernter's geometry-program did not cheat -- not merely because it was too rigid to cheat in any way, but also because it could not have cheated in this way. It knew nothing of the third dimension. Indeed, it had no visual, analogical, representation of triangles at all. It represented a triangle not as a two-dimensional spatial form, but as a list of three letters (e.g. ABC) naming points in an abstract coordinate space. Similarly, it represented an angle as a list of three letters naming the vertex and one of the points on each of the two rays. Being unable to inspect triangles visually, it even had to prove that every different letter-name for what we can see to be the same angle was equivalent. So it had to prove (for instance) that angle XYZ is the same as angle ZYX, and angle BAC the same as angle CAB. Consequently, this program was incapable not only of coming up with Pappus' proof in the way he did, but even of representing such a proof -- or of appreciating its elegance and originality. Its mental maps simply did not allow for the lifting and replacement of triangles in space (and it had no heuristics enabling it to transform those maps).

How did it come up with its pseudo-Pappus proof, then? Treating the "ABC's" as (spatially uninterpreted) abstract vectors, it did a massive brute-search to find the proof. Since this brute search succeeded, it did not bother to construct any extra lines.

This example shows how careful one must be in ascribing creativity to a person, and in answering the second Lovelace question about a program. We have to consider not only the resulting idea, but also the mental processes which gave rise to it. Brute force search is even less creative than associative (improbabilist) thinking, and problem-dimensions which can be mapped by some systems may not be representable by others. (Analogously, a three-year old not showing flexible imagination in drawing a funny man: rather, she is showing incompetence in drawing an ordinary man.)

It should not be assumed from the example of Pappus (or Kekule) that visual imagery is always useful in mapping and transforming one's ideas. An example is given of a problem for which a visual representation is almost always constructed, but which hinders solution. Where mental maps are concerned, visual maps are not always best.

Chapter 6: Creative Connections

This chapter deals with associative creativity: the spontaneous generation of new ideas, and/or novel combinations of familiar ideas, by means of unconscious processes of association. Examples include not only "mere associations" but also analogies, which may then be consciously developed for purposes of rhetorical exposition or problem-solving. In Chapter 6, I discuss the initial association of ideas. (The evaluation and use of analogy are addressed in Chapter 7.)

One of the richest veins of associative creativity is poetic imagery. I consider some specific examples taken from Coleridge's poem The Ancient Mariner. For this poem (and also for his Kubla Khan), we have unusually detailed information about the literary sources of the imagery concerned. The literary scholar John Livingston Lowes (1951) studied Coleridge's Notebooks written while preparing for and writing the poem, and followed up every source mentioned there -- and every footnote given in each source. Despite the enormous quantity and range of Coleridge's reading, Lowes makes a subtle, and intuitively compelling, case in identifying specific sources for the many images in the poem.

However, an intuitively compelling case is one thing, and an explicit justification or detailed explanation is another. Lowes took for granted that association can happen (he used Coleridge's term: the hooks and eyes of memory), without being able to say just how these hooks and eyes can come together. I argue that connectionism, and specifically PDP (parallel distributed processing), can help us to understand how such unexpected associations are possible.

Among the relevant questions to which PDP-models offer preliminary answers are the following: How can ideas from very different sources (such as Captain Cook's diaries and Priestley's writings on optics) be spontaneously thought of together? How can two ideas be merged to produce a new structure, which shows the influence of both ancestor-ideas without being a mere "cut-and-paste" combination? How can the mind be "primed" (for instance, by the decision to write a poem about a seaman), so that one will more easily notice serendipitous ideas? Why may someone notice -- and remember -- something fairly uninteresting (such as a word in a literary text), if it occurs in an interesting context? How can a brief phrase conjure up from memory an entire line or stanza, from this or some other poem? And how can we accept two ideas as similar (the words "love" and "prove" as rhyming, for instance) in respect of a feature not identical in both?

The features of connectionist models which suggest answers to these questions are their powers of pattern-completion, graceful degradation, sensitization, multiple constraint-satisfaction, and "best-fit" equilibration. The computational processes underlying these features are described informally in Chapter 6 (I assume that it is not necessary to do so for BBS-readers).

The message of this chapter is that the unconscious, "insightful", associative aspects of creativity can be explained -- in outline, at least -- in computational terms. Connectionism offers some specific suggestions about what sorts of processes may underlie the hooks and eyes of memory.

This is not to say, however, that all aspects of poetry -- or even all poetic imagery -- can be explained in this way. Quite apart from the hierarchical structure of natural language itself, some features of a poem may require thinking of a type more suited (at present) to symbolic models. For example, Coleridge's use of "The Sun came up upon the left" and "The Sun now rose upon the right" as the opening-lines of two closely-situated stanzas enabled him to indicate to the reader that the ship was circumnavigating the globe, without having to detail all the uneventful miles of the voyage. (Compare Kubrick's use of the spinning thigh-bone turning into a space-ship, as a highly compressed history of technology, in his film 2001, A Space Odyssey.) But these expressions, too, were drawn from his reading -- in this case, of the diaries of the very early mariners, who recorded their amazement at first experiencing the sunrise in the "wrong" part of the sky. Associative memory was thus involved in this poetic conceit, but it is not the entire explanation.

Chapter 7: Unromantic Artists

This chapter and the next describe and criticize some existing computer models of creativity. The separation into "artists" (Chapter 7) and "scientists" (Chapter 8) is to some extent an arbitrary rhetorical device. For example, analogy (discussed in Chapter 7) and induction and genetic algorithms (both outlined in Chapter 8) are all relevant to creativity in arts and sciences alike. In these two chapters, the second and third Lovelace-questions -- about apparent computer-creativity -- are addressed at length. However, the first Lovelace question, relating to human creativity, is still the over-riding concern.

The computer models of creativity discussed in Chapter 7 include: a series of programs which produce line-drawings (McCorduck, 1991); a jazz-improviser (Johnson-Laird, 1991); a haiku-writer (Masterman & McKinnon Wood, 1968); two programs for writing stories (Klein et al., 1973; Meehan, 1981); and two analogy-programs (Chalmers, French, & Hofstadter, 1991; Holyoak & Thagard, 1989a, 1989b; Mitchell, 1993). In each case, the programmer has to try to define the dimensions of the relevant conceptual space, and to specify ways of exploring the space, so as to generate novel structures within it. Some evaluation, too, must be allowed for. In the systems described in this chapter, the evaluation is built into the generative procedures, rather than being done post hoc. (This is not entirely unrealistic: although humans can evaluate -- and modify -- their own ideas once they have produced them, they can also develop domain-expertise such that most of their ideas are acceptable without modification.)

Sometimes, the results are comparable with non-trivial human achievements. Thus some of the computer's line-drawings are spontaneously admired, by people who are amazed when told their provenance. The haiku-program can produce acceptable poems, sometimes indistinguishable from human-generated examples (however, this is due to the fact that the minimalist haiku-style demands considerable projective interpretation by the reader). And the jazz-program can play -- composing its own chord-sequences, as well as improvising on them -- at about the level of a moderately competent human beginner. (Another jazz-improviser, not mentioned in the book, plays at the level of a mediocre professional musician; unlike the former example, it starts out with significant musical structure provided to it "for free" by the human user (Hodgson, 1990).)

At other times, the results are clumsy and unconvincing, involving infelicities and absurdities of various kinds. This often happens when stories are computer-generated. Here, many rich conceptual spaces have to be negotiated simultaneously. Quite apart from the challenge of natural language generation, the model must produce sensible plots, taking account both of the motivation and action of the characters and of their common-sense knowledge. Where very simple plot-spaces, and very limited world-knowledge, are concerned, a program may be able (sometimes) to generate plausible stories.

One, for example, produces Aesop-like tales, including a version of "The Fox and the Crow" (Meehan, 1981). A recent modification of this program (Turner, 1992), not covered in the book, is more subtle. It uses case-based reasoning and case-transforming heuristics to generate novel stories based on familiar ones; and because it distinguishes the author's goals from those of the characters, it can solve meta-problems about the story as well as problems posed within it. But even this model's story-telling powers are strictly limited, compared with ours.

Models dealing with the interpretation of stories, and of concepts (such as betrayal) used in stories, are also relevant here. Computational definitions of interpersonal themes and scripts (Abelson, 1973), programs that can answer questions about (simple) stories and models which can -- up to a point -- interpret motivational and emotional structures within a story (Dyer, 1983) are all discussed.

So, too, is a program that generates English text describing games of noughts-and-crosses (Davey, 1978). The complex syntax of the sentences is nicely appropriate to the structure of the particular game being described. Human writers, too, often use subtleties of syntax to convey certain aspects of their story-lines.

The analogy programs described in Chapter 7 are ACME and ARCS (Holyoak & Thagard, 1989a, 1989b), and in the Preface to the paperback edition I add a discussion of Copycat (Chalmers et al., 1991; Mitchell, 1993), which I had originally intended to highlight in the main text.

ACME and ARCS are an analogy-interpreter and an analogy-finder, respectively. Calling on a semantic net of over 30,000 items, to which items can be added by the user, these programs use structural, semantic, and pragmatic criteria to evaluate analogies between concepts (whose structure is pre-given by the programmers). Other analogy programs (e.g. Falkenhainer, Forbus, & Gentner, 1989) use structural and semantic similarity as criteria. But ARCS/ACME takes account also of the pragmatic context, the purpose for which the analogy is being sought. So a conceptual feature may be highlighted in one context, and downplayed in another. The context may be one of rhetoric or poetic imagery, or one of scientific problem-solving (ARCS/ACME forms part of an inductive program that compares the "explanatory coherence" of rival scientific theories (Thagard, 1992)). Examples of both types are discussed.

The point of interest about Copycat is that it is a model of analogy in which the structure of the analogues is neither pre-assigned nor inflexible. The description of something can change as the system searches for an analogy to it, and its "perception" of an analogue may be permanently influenced by having seen it in a particular analogical relation to something else. Many analogies in the arts and sciences can be cited, to show that the same is true of the human mind.

Among the points of general interest raised in this chapter is the inability of these programs (Copycat excepted) to reflect on what they have done, or to change their way of doing it.

For instance, the line-drawing program that draws human acrobats in broadly realistic poses is unable to draw one-armed acrobats. It can generate acrobats with only one arm visible, if one arm is occluded by another acrobat in front. But that there might be a one-armed (or a six-armed) acrobat is strictly inconceivable. The reason is that the program's knowledge of human anatomy does not represent the fact that humans have two arms in a form which is separable from its drawing-procedures or modifiable by "imaginative" heuristics. It does not, for instance, contain anything of the form "Number of arms: 2", which might then be transformed by a "vary the variable" heuristic into "Number of arms: 1". Much as the four-year-old child cannot draw a "funny" one-armed man because she has not yet developed the necessary RR of her own man-drawing skill, so this program cannot vary what it does because -- in a clear sense -- it does not know what it is that it is doing.

This failing is not shared by all current programs: some featured in the next chapter can evaluate their own ideas, and transform their own procedures, to some extent. Moreover, this failure is "bad news" only to those seeking a positive answer to the second and third Lovelace questions. It is useful to anyone asking the first Lovelace question, for it underlines the importance of the factors introduced in Chapter 4: reflexive mapping of thought, evaluation of ideas, and transformation of conceptual spaces.

Chapter 8: Computer-Scientists

Like analogy, inductive thinking occurs across both arts and science. Chapter 8 begins with a discussion of the ID3 algorithm. This is used in many learning programs, including a world-beater -- better than the human expert who "taught" it -- at diagnosing soybean diseases (Michalski & Chilausky, 1980).

ID3 learns from examples. It looks for the logical regularities which underlie the classification of the input examples, and uses them to classify new, unexamined, examples. Sometimes, it finds regularities of which the human experts were unaware, such as unknown strategies for chess endgames (Michie & Johnston, 1984). In short, ID3 can not only define familiar concepts in H-creative ways, but can also define H-creative concepts.

However, all the domain-properties it considers have to be specifically mentioned in the input. (It does not have to be told just which input properties are relevant: in the chess end-game example, the chess-masters "instructing" the program did not know this.) That is, ID3-programs can restructure their conceptual space in P-creative -- and even H-creative -- ways. But they cannot change the dimensions of the space, so as to alter its fundamental nature.

Another program capable of H-discovery is meta-DENDRAL, an early expert system devoted to the spectroscopic analysis of a certain group of organic molecules. The original program, DENDRAL, uses exhaustive search to describe all possible molecules made up of a given set of atoms, and heuristics to suggest which of these might be chemically interesting. DENDRAL uses only the chemical rules supplied to it, but meta-DENDRAL can find new rules about how these compounds decompose. It does this by identifying unfamiliar patterns in the spectrographs of familiar compounds, and suggesting plausible explanations for them. For instance, if it discovers a smaller structure located near the point at which a molecule breaks, it may suggest that other molecules containing that sub-structure may break at these points too.

This program is H-creative, up to a point. It not only explores its conceptual space (using evaluative heuristics and exhaustive search) but enlarges it too, by adding new rules. It generates hunches, which have led to the synthesis of novel, chemically interesting, compounds. And it has discovered some previously unsuspected rules for analysing several families of organic molecules. However it relies on sophisticated theories built into it by expert chemists (which is why its novel hypotheses, though sometimes false, are always plausible). It casts no light on how those theories might have arisen in the first place.

Some computational models of induction were developed with an eye to the history of science (and to psychology), rather than for practical scientific puzzle-solving. Their aim was not to come up with H-creative ideas, but to P-create in the same way as human H-creators. Examples include BACON, GLAUBER, STAHL, and DALTON (Langley, Simon, Bradshaw, & Zytkow, 1987), whose P-creative activities are modelled on H-creative episodes recorded in the notebooks of human scientists.

BACON induces quantitative laws from empirical data. Its data are measurements of various properties at different times. It looks for simple mathematical functions defining invariant relations between numerical data-sets. For instance, it seeks direct or inverse proportionalities between measurements, or between their products or ratios. It can define higher-level theoretical terms, construct new units of measurement, and use mathematical symmetry to help find invariant patterns in the data. It can cope with noisy data, finding a best-fit function (within predefined limits). BACON has P-created many physical laws, including Archimedes' principle, Kepler's third law, Boyle's law, Ohm's law, and Black's law.

GLAUBER discovers qualitative laws, summarizing the data by classifying things according to (non-measurable) observable properties. Thus it discovers relations between acids, alkalis, and bases (all identified in qualitative terms). STAHL analyses chemical compounds into their elements. Relying on the data-categories presented to it, it has modelled aspects of the historical progression from phlogiston-theory to oxygen-theory. DALTON reasons about atoms and molecular structure. Using early atomic theory, it generates plausible molecular structures for a given set of components (it could be extended to cover other componential theories, such as particle physics or Mendelian genetics).

These four programs have rediscovered many scientific laws. However, their P-creativity is shallow. They are highly data-driven, their discoveries lying close to the evidence. They cannot identify relevance for themselves, but are "primed" with appropriate expectations. (BACON expects to find linear relationships, and rediscovered Archimedes' principle only after being told that things can be immersed in known volumes of liquid and the resulting volume measured.) They cannot model spontaneous associations or analogies, only deliberate reasoning. Some can suggest experiments, to test hypotheses they have P-created, but they have no sense of the practices involved. They can learn, constructing P-novel concepts used to make further P-discoveries. But their discoveries are exploratory rather than transformational: they cannot fundamentally alter their own conceptual spaces.

Some AI-models of creativity can do this, to some extent. For instance, the Automatic Mathematician (AM) explores and transforms mathematical ideas (Lenat, 1983). It does not prove theorems, or do sums, but generates "interesting" mathematical ideas (including expressions that might be provable theorems). It starts with 100 primitive concepts of set-theory (such as set, list, equality, and ordered pair), and 300 heuristics that can examine, combine, transform, and evaluate its concepts. One generates the inverse of a function (compare "consider the negative"). Others can compare, generalize, specialize, or find examples of concepts. Newly-constructed concepts are fed back into the pool.

In effect, AM has hunches: its evaluation heuristics suggest which new structures it should concentrate on. For example, AM finds it interesting whenever the union of two sets has a simply expressible property which is not possessed by either of them (a set-theoretic version of the notion that emergent properties are interesting). Its value-judgments are often wrong. Nevertheless, it has constructed some powerful mathematical notions, including prime numbers, Goldbach's conjecture, and an H-novel theorem concerning maximally-divisible numbers (which the programmer had never heard of). In short, AM appears to be significantly P-creative, and slightly H-creative too.

However, AM has been criticised (Haase, 1986; Lenat & Seely-Brown, 1984; Ritchie & Hanna, 1984; Rowe & Partridge, 1993). Critics have argued that some heuristics were included to make certain discoveries, such as prime numbers, possible; that the use of LISP provided AM with mathematical relevance "for free", since any syntactic change in a LISP expression is likely to result in a mathematically-meaningful string; that the program's exploration was too often guided by the human user; and that AM had fixed criteria of interest, being unable to adapt its values. The precise extent of AM's creativity, then, is unclear.

Because EURISKO has heuristics for changing heuristics, it can transform not only its stock of concepts but also its own processing-style. For example, one heuristic asks whether a rule has ever led to any interesting result. If it has not (but has been used several times), it will be less often used in future. If it has occasionally been helpful, though usually worthless, it may be specialized in one of several different ways. (Because it is sometimes useful and sometimes not, the specializing-heuristic can be applied to itself.) Other heuristics generalize rules, or create new rules by analogy with old ones. Using domain-specific heuristics to complement these general ones, EURISKO has generated H-novel ideas in genetic engineering and VLSI-design (one has been patented, so was not "obvious to a person skilled in the art").

Other self-transforming systems described in this chapter are problem-solving programs based on genetic algorithms (GAs). GA-systems have two main features. They all use rule-changing algorithms (mutation and crossover) modelled on biological genetics. Mutation makes a random change in a single rule. Crossover mixes two rules, so that (for instance) the lefthand portion of one is combined with the righthand portion of the other; the break-points may be chosen randomly, or may reflect the system's sense of which rule-parts are the most useful. Most GA-systems also include algorithms for identifying the relatively successful rules, and rule-parts, and for increasing the probability that they will be selected for "breeding" future generations. Together, these algorithms generate a new system, better adapted to the task.

An example cited in the book is an early GA-program which developed a set of rules to regulate the transmission of gas through a pipeline (Holland, Holyoak, Nesbitt, & Thagard, 1986). Its data were hourly measurements of inflow, outflow, inlet-pressure, outlet-pressure, rate of pressure-change, season, time, date, and temperature. It altered the inlet-pressure to allow for variations in demand, and inferred the existence of accidental leaks in the pipeline (adjusting the inflow accordingly).

Although the pipeline-program discovered the rules for itself, the potentially relevant data-types were given in its original list of concepts. How far that compromises its creativity is a matter of judgment. No system can work from a tabula rasa. Likewise, the selectional criteria were defined by the programmer, and do not alter. Humans may be taught evaluative criteria, too. But they can sometimes learn -- and adapt -- them for themselves.

GAs, or randomizing thinking, are potentially relevant to art as well as to science -- especially if the evaluation is done interactively, not automatically. That is, at each generation the selection of items from which to breed for the next generation is done by a human being. This methodology is well-suited to art, where the evaluative criteria are not only controversial but also imprecise -- or even unknown. Two recent examples (not mentioned in the book, but described in: Boden, in press) concern graphics (Sims, 1991; Todd & Latham, 1993). Sims' aim is to provide an interactive environment for graphic artists, enabling them to generate otherwise unimaginable images. Latham's is to produce his own art-works, but he too uses the computer to generate images he could not have developed unaided.

In a run of Sims' GA-system, the first image is generated at random. Then the program makes various independent random mutations in the image-generating rule, and displays the resulting images. The human now chooses one image to be mutated, or two to be "mated", and the process is repeated. The program can transform its image-generating code (simple LISP-functions) in many ways. It can alter parameters in pre-existing functions, combine or separate functions, or nest one function inside another (so many-levelled hierarchies can arise).

Many of Sims' computer-generated images are highly attractive, even beautiful. Moreover, they often cause a deep surprise. The change(s) between parent and offspring are sometimes amazing. The one appears to be a radical transformation of the other -- or even something entirely different. In short, we seem to have an example of impossibilist creativity.

Latham's interactive GA-program is much more predictable. Its mutation operators can change only the parameters within the image-generating code, not the body of the function. Consequently, it never comes up with radical novelties. All the offspring in a given generation are obviously siblings, and obviously related to their parents. So the results of Latham's system are less exciting than Sims'. But it is arguably even more relevant to artistic creativity.

The interesting comparison is not between the aesthetic appeal of a typical Latham-image and Sims-image, but between the discipline -- or lack of it -- which guides the exploration and transformation of the relevant visual space. Sims is not aiming for particular types of result, so his images can be fundamentally transformed in random ways at every generation. But Latham (a professional artist) has a sense of what forms he hopes to achieve, and specific aesthetic criteria for evaluating intermediate steps. Random changes at the margins are exploratory, and may provide some useful ideas. But fundamental transformations -- especially, random ones -- would be counterproductive. (If they were allowed, Latham would want to pick one and then explore its possibilities in a disciplined way.)

This fits the account of (impossibilist) creativity given in Chapters 3 and 4. Creativity works within constraints, which define the conceptual spaces with respect to which it is identified. Maps or RRs (or LISP-functions) which describe the parameters and/or the major dimensions of the space can be altered in specific ways, to generate new, but related, spaces.

Random changes are sometimes helpful, but only if they are integrated into the relevant style. Art, like science, involves discipline. Only after a space has been fairly thoroughly explored will the artist want to transform it in deeply surprising ways. A convincing computer-artist would therefore need not only randomizing operators, but also heuristics for constraining its transformations and selections in an aesthetically acceptable fashion. In addition, it would need to make its aesthetic selections (and perhaps guiding recommendations) for itself. And, to be true to human creativity, the evaluative rules should evolve also (Elton, 1993).

Chapter 9: Chance, Chaos, Randomness, Unpredictability

Unpredictability is often said to be the essence of creativity. And creativity is, by definition, surprising. But unpredictability is not enough. At the heart of creativity, as previous chapters have shown, lie constraints: the very opposite of unpredictability. Constraints and unpredictability, familiarity and surprise, are somehow combined in original thinking.

In this chapter, I distinguish various senses of "chance", "chaos", "randomness", and "unpredictability". I also argue that a scientific explanation need not imply either determinism or predictability, and that even deterministic systems may be unpredictable. Below, it will suffice to mention a number of different ways in which unpredictability can enter into creativity.

The first follows from the fact that creative constraints do not determine everything about the newly-generated idea. A style of thinking typically allows for many points at which two or more alternatives are possible. Several notes may be both melodious and harmonious; many words rhyme with moon; and perhaps there could be a ring-molecule with three, or five, atoms in the ring? At these points, some specific choice must be made. Likewise, many exploratory and transformational heuristics may be potentially available at a certain time, in dealing with a given conceptual space. But one or other must be chosen. Even if several heuristics can be applied at once (like parallel mutations in a GA-system), not all possibilities can be simultaneously explored. The choice has to be made, somehow.

Occasionally, the choice is random, or as near to random as one can get. So it may be made by throwing a dice (as in playing Mozart's aleatory music); or by consulting a table of random numbers (as in the jazz-program); or even, possibly, as a result of some sudden quantum-jump inside the brain. There may even be psychological processes akin to GA-mechanisms, producing novel ideas in human minds.

More often, the choice is fully determined, by something which bears no systematic relation to the conceptual space concerned. (Some examples are given below.) Relative to that style of thinking, the choice is made randomly. Certainly, nothing within the style itself could enable us to predict its occurrence.

In either case, the choice must somehow be skilfully integrated into the relevant mental structure. Without such disciplined integration, it cannot lead to a positively valued, interesting, idea. With the help of this mental discipline, even flaws and accidents may be put to creative use. For instance, a jazz-drummer suffering from Tourette's syndrome is subject to sudden, uncontrollable, muscular tics, even when he is drumming. As a result, his drumsticks sometimes make unexpected sounds. But his musical skill is so great that he can work these supererogatory sounds into his music as he goes along. At worst, he "covers up" for them. At best, he makes them the seeds of unusual improvisations which he could not otherwise have thought of.

One might even call the drummer's tics serendipitous. Serendipity is the unexpected finding of something one was not specifically looking for. But the "something" has to be something which was wanted, or at least which can now be used. Fleming's discovery of the dirty petri-dish, infected by Penicillium spores, excited him because he already knew how useful a bactericidal agent would be. Proust's madeleine did not answer any currently pressing question, but it aroused a flood of memories which he was able to use as the trigger of a life-long project. Events such as these could not have been foreseen. Both trigger and triggering were unpredictable. Who was to say that the dish would be left uncovered, and infected by that particular organism? And who could say that Proust would eat a madeleine on that occasion? Even if one could do this (perhaps the laboratory was always untidy, and perhaps Proust was addicted to madeleines), one could not predict the effect the trigger would have on these individual minds.

This is so even if there are no absolutely random events going on in our brains. Chaos theory has taught us that fully deterministic systems can be, in practice, unpredictable. Our inescapable ignorance of the initial conditions means that we cannot forecast the weather, except in highly general (and short-term) ways. The inner dynamics of the mind are more complex than those of the weather, and the initial conditions -- each person's individual experiences, values, and beliefs -- are even more varied. Small wonder, then, if we cannot fully foresee the clouds of creativity in people's minds.

To some extent, however, we can. Different thinkers have differing individual styles, which set a characteristic stamp on all their work in a given domain. Thus Dr. Johnson complained, "Who but Donne would have compared a good man to a telescope?". Authorial signatures are largely due to the fact that people can employ habitual ways of making "random" choices. There may be nothing to say, beforehand, how someone will choose to play the relevant game. But after several years of practice, their "random" choices may be as predictable as anything in the basic genre concerned.

More mundane examples of creativity, which are P-creative but not H-creative, can sometimes be predicted -- and even deliberately brought about. Suppose your daughter is having difficulty mastering an unfamiliar principle in her physics homework. You might fetch a gadget that embodies the principle concerned, and leave it on the kitchen-table, hoping that she will play around with it and realise the connection for herself. Even if you have to drop a few hints, the likelihood is that she will create the central idea. Again, Socratic dialogue helps people to explore their conceptual spaces in (to them) unexpected ways. But Socrates himself, like those taking his role today, knew what P-creative ideas to expect from his pupils.

We cannot predict creative ideas in detail, and we never shall be able to do so. Human experience is too richly idiosyncratic. But this does not mean that creativity is fundamentally mysterious, or beyond scientific understanding.

Chapter 10: Elite or Everyman?

Creativity is not a single capacity, and nor is it a special one. It is an aspect of intelligence in general, which involves many different capacities: noticing, remembering, seeing, speaking, classifying, associating, comparing, evaluating, introspecting, and the like. Chapter 10 offers evidence for this view, drawing on the work of Perkins (1981) and also on computational work of various kinds.

For example, Kekule's description of "long rows, twining and twisting in snakelike motion", where "one of the snakes had seized hold of its own tail", assumes everyday powers of visual interpretation and analogy. These capacities are normally taken for granted in discussions of Kekule's H-creativity, but they require some psychological explanation. Relevant computational work on low-level vision suggests that Kekule's imagery was grounded in certain specific, and universal, visual capacities -- including the ability to identify lines and end-points. (His hunch, by contrast, required special expertise. As remarked in Chapter 4, only a chemist could have realized the potential significance of the change in neighbour-relations caused by the coalescence of end-points, or the "snake" which "seized hold of its tail".)

Similarly, Mozart's renowned musical memory, and his reported capacity for hearing a whole symphony "all at once", can be related to computational accounts of powers of memory and comprehension common to us all. Certainly, his musical expertise was superior in many ways. He had a better grasp of the conceptual spaces concerned, and a better understanding -- better even than Salieri's -- of how to explore them so as to locate their farthest nooks and crannies. (Unlike Haydn, for example, he was not a composer who made adventurous transformations). But much of Mozart's genius may have lain in the better use, and the vastly more extended practice, of facilities we all share.

Much -- but perhaps not all. Possibly, there was something special about Mozart's brain which predisposed him to musical genius (Gardner, 1983). However, we have little notion, at present, of what this could be. It may have been some cerebral detail which had the emergent effect of giving him greater musical powers. For example, the jazz-improvisation program described in Chapter 7 employed only very simple rules to improvise, because its short-term memory was deliberately constrained to match the limited STM of people. Human jazz-musicians cannot improvise hierarchically nested chord-sequences "on the fly", but have to compose (or memorize) them beforetimes. A change in the range of STM might enable someone to improvise and appreciate musical structures of a complexity not otherwise intelligible. But this musically significant change might be due to an apparently "boring" feature of the brain.

Many other examples of creativity (drawn, for instance, from poetry, painting, music, and choreography) are cited in this chapter. They all rely on familiar capacities for their effect, and arguably for their occurrence too. We appreciate them intuitively, and normally take their accessibility -- and their origins -- for granted. But psychological explanations in computational terms may be available, at least in outline.

The role of motivation and emotion is briefly mentioned, but is not a prime theme. This is not because motivation and emotion are in principle outside the reach of a computational psychology. Some attempts have been made to bring these matters within a computational account of the mind (e.g. Boden, 1972; Sloman, 1987). But such attempts provide outline sketches rather than functioning models. Still less is it because motivation is irrelevant to creativity. But the main topic of the book is how (not why) novel ideas arise in human minds.

Chapter 11: Of Humans and Hoverflies

The final chapter focusses on two questions. One is the fourth Lovelace question: could a computer really be creative? The other is whether any scientific explanation of creativity, whether computational or not, would be dehumanizing in the sense of destroying our wonder at it -- and at the human mind in general.

With respect to the fourth Lovelace question, the answer "No" may be defended in at least four different ways. I call these the brain-stuff argument, the empty-program argument, the consciousness argument, and the non-human argument. Each of these applies to intelligence (and intentionality) in general, not just to creativity in particular.

The brain-stuff argument (Searle, 1980) claims that whereas neuroprotein is a kind of stuff which can support intelligence, metal and silicon are not. This empirical claim is conceivably correct, but we have no specific reason to believe it. Moreover, the associated claim -- that it is intuitively obvious that neuroprotein can support intentionality and that metal and silicon cannot -- must be rejected.

Intuitively speaking, that neuroprotein supports intelligence is utterly mysterious: how could that grey mushy stuff inside our skulls have anything to do with intentionality? Insofar as we understand this, we do so because of various functions that nervous tissue makes possible (as the sodium pump enables action potentials, or "messages", to pass along an axon). Any material substrate capable of supporting all the relevant functions could act as the embodiment of mind. Whether neurochemistry describes the only such substrate is an empirical question, not to be settled by intuitions.

The empty-program argument is Searle's (1980) claim that a computational psychology cannot explain understanding, because programs are all syntax and no semantics: their symbols are utterly meaningless to the computer itself. I reply that a computer program, when running in a computer, has proto-semantic (causal) properties, in virtue of which the computer does things -- some of which are among the sorts of thing which enable understanding in humans and animals (Boden, 1988, ch. 8; Sloman, 1986). (This is not to say that any computer-artefact could possess understanding in the full sense, or what I have termed "intrinsic interests", grounded in evolutionary history (Boden, 1972).)

The consciousness argument is that no computer could be conscious, and therefore -- since consciousness is needed for the evaluation phase, and even for much of the preparation phase -- no computer can be creative. I reply that it's not obvious that evaluation must be carried out consciously. A creative computer might recognize (evaluate) its creative ideas by using relevant reflexive criteria without also having consciousness. Moreover, some aspects of consciousness can be illuminated by a computational account, although admittedly "qualia" present an unsolved problem. The question must remain open -- not just because we do not know the answer, but because we do not clearly understand how to ask the question.

According to the non-human argument, to regard computers as truly intelligent is not a mere factual mistake, but a moral absurdity: only members of the human, or animal, community should be granted moral and epistemological consideration (of their interests and opinions). If we ever agreed to remove all the scare-quotes around the psychological words we use in describing computers, so inviting them to join our human community, we would be committed to respecting their goals and judgments. This would not be a purely factual matter, but one of moral and political choice -- about which it is impossible to legislate now.

In short, each of the four negative replies to the last Lovelace question is challengeable. But even someone who does accept a negative answer here can consistently accept positive answers to the first three Lovelace questions. The main argument of the book remains unaffected.

The second theme of this final chapter is the question whether, where creativity is in question, scientific explanation in general should be spurned. Many people, from Blake to Roszak, have seen the natural sciences as dehumanizing in various ways. Three are relevant here: the ignoring of mentalistic concepts, the denial of cherished beliefs, and the destructive demystification of some valued phenomena.

The natural sciences have had nothing to say about psychological phenomena as such; and scientifically-minded psychologists have often conceptualized them in reductionist (e.g. behaviourist, or physiological) terms. To ignore something is not necessarily to deny it. But, given the high status of the natural sciences, the fact that they have not dealt with the mind has insidiously downplayed its importance, if not its very existence.

This charge cannot be levelled at computational psychology, however. Intentional concepts, such as representation, lie at the heart of it, and of AI. Some philosophers claim that these sciences have no right to use such terms. Even so, they cannot be accused of deliberately ignoring intentional phenomena, or of rejecting intentionalist vocabulary.

The second charge of dehumanization concerns what science explicitly denies. Some scientific theories have rejected comforting beliefs, such as geocentrism, special creation, or rational self-control. But a scientific psychology need not -- and a computational psychology does not -- deny creativity, as astronomy denies geocentrism. On the contrary, the preceding chapters have acknowledged creativity again and again. Even to say that it rests on universal features of human minds is not to deny that some ideas are surprising, and special, requiring explanation of how they could possibly arise.

However, the humanist's worry concerns not only denial by rejection, but also denial by explanation. The crux of the third type of anti-scientific resistance is the feeling that scientific explanation of any kind must drive out wonder: that to explain something is to cease to marvel at it. Not only do we wonder at creativity, but positive evaluation is essential to the concept. So it may seem that to explain creativity is insidiously to downgrade it -- in effect, to deny it.

Certainly, many examples can be given where understanding drives out wonder. For instance, we may marvel at the power of the hoverfly to fly to its mate hovering nearby (so as to mate in mid-air). Many people might be tempted to describe the hoverfly's activities in terms of its goals and beliefs, and perhaps even its determination in going straight to its mate without any coyness or prevarication. How wonderful is the mind of the humble hoverfly!

In fact, the hoverfly's flight-path is determined by a simple and inflexible rule, hardwired into its brain. This rule transforms a specific visual signal into a specific muscular response. The fly's initial change of direction depends on the particular approach-angle subtended by the target-fly. The creature, in effect, always assumes that the size and velocity of the seen target (which may or may not be a fly) are those corresponding to hoverflies. When initiating a new flight-path, the fly's angle of turn is selected on this rigid, and fallible, basis. Moreover, the fly's path cannot be adjusted in midflight, there being no way in which it can be influenced by feedback from the movement of the target animal.

This evidence must dampen the enthusiasm of anyone who had marvelled at the psychological subtlety of the hoverfly's behaviour. The insect's intelligence has been demystified with a vengeance, and it no longer seems worthy of much respect. One may see beauty in the evolutionary principles that enabled this simple computational mechanism to develop, or in the biochemistry that makes it function. But the fly itself cannot properly be described in anthropomorphic terms. Even if we wonder at evolution, and at insect-neurophysiology, we can no longer wonder at the subtle mind of the hoverfly.

Many people fear that this disillusioned denial of intelligence in the hoverfly is a foretaste of what science will say about our minds too. A few "worrying" examples can indeed be given: for instance, think of how perceived sexual attractiveness turns out to relate to pupil-size. In general, however, this fear is mistaken. The mind of the hoverfly is much less marvellous than we had imagined, so our previous respect for the insect's intellectual prowess is shown up as mere ignorant sentimentality. But computational explanations of thinking can increase our respect for human minds, by showing them to be much more complex and subtle than we had previously recognized.

Consider, for instance, the many different ways (some are sketched in Chapters 4 and 5) in which Kekule could have seen snakes as suggesting ring-molecules. Think of the rich analogy-mapping in Coleridge's mind, which drew on naval memoirs, travellers' tales, and scientific reports to generate the imagery of The Ancient Mariner (Chapter 6). Bear in mind the mental complexities (outlined in Chapter 7) of generating an elegant story-line, or improvising a jazz-melody. And remember the many ways in which random events (the mutations described in Chapter 8, or the serendipities cited in Chapter 9) may be integrated into pre-existing conceptual spaces with creative effect.

Writing about Coleridge's imagery, Livingston Lowes said: "I am not forgetting beauty. It is because the worth of beauty is transcendent that the subtle ways of the power that achieves it are transcendently worth searching out." His words apply not only to literary studies of creativity, but to scientific enquiry too. A scientific psychology, whether computational or not, allows us plenty of room to wonder at Mozart, or at our friends' jokes. Psychology leaves poetry in place. Indeed, it adds a new dimension to our awe on encountering creative ideas, for it helps us to see the richness, and yet the discipline, of the underlying mental processes.

To understand, even to demystify, is not necessarily to denigrate. A scientific explanation of creativity shows how extraordinary is the ordinary person's mind. We are, after all, humans -- not hoverflies.

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