Archimedes leapt from his bath shouting “Eureka!” after discovering how to measure the volume of irregular objects. Kekulé, while daydreaming, envisioned benzene’s ring structure. Stories like these are a common experience of creativity. Creative ideas mysteriously emerge as flashes of insight from the unconscious mind.

Yet the mystery runs even deeper. Margaret Boden notes: “How it happens is indeed puzzling, but that it happens at all is deeply mysterious.” How can something genuinely novel arise seemingly from nowhere? Why are humans capable of creating new ideas at all? Historically, this led many to see creativity as a divine gift granted to a lucky few, placing it seemingly beyond the reach of scientific explanation. But Boden challenges this view, arguing instead that creativity is a fundamental cognitive ability shared by all humans, which can be scientifically understood and even modeled by machines.

When Margaret Boden first published "The Creative Mind," she already exemplified creativity within machines, although only a few and in a limited form. Now, three decades later, we have reached a moment where machines—particularly large language models (LLMs)—are arriving at impressive products of creativity. In this article, we revisit Boden's influential framework, then exemplify it through Kekulé’s iconic insight, and finally examine current AI creativity through the framework.

What can Boden teach us about creativity?

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Boden offers a more democratic description of creativity compared to historical views, seeing it as a fundamental cognitive process inherent in everyone and capable of being nurtured. She defines creativity as “the ability to come up with ideas or artefacts that are new, surprising and valuable.” Ideas include abstract creations such as poems, musical compositions, scientific theories, recipes, and jokes. Artifacts refer to tangible creations like paintings, sculptures, steam engines, pottery, and origami. Therefore, creativity permeates all aspects of human activity.

At the core of creativity, there are three key aspects. The first aspect—being new—has two interpretations. Boden distinguishes between Psychological novelty, referring to ideas that are new to the individual, and Historical novelty, which typically gains recognition and enters the history books. In Boden’s framework, the focus is on the psychological kind because the interest is on the cognitive processes of how an individual might generate a thought they have never previously experienced.

We saw that "new" has two meanings, and we'll see that "surprising" has three. As for "valuable," there can be as many meanings as there are judgment criteria - something can be valuable because it is beautiful, elegant, useful, performs well on a benchmark, or simply makes your grandma happy. We won't explore this aspect much further, as we'll focus instead on the final one.

In Boden’s work, the most detailed aspect is that of being surprising, which essentially means challenging our expectations. Boden suggests three distinct ways in which creative outputs can do that: combinational, exploratory, and transformative. These can be viewed as existing along a continuum, each representing an increasing degree of surprise or expectations being challenged.

Combinational creativity involves making unfamiliar combinations of familiar ideas. A good example is political satire, where famous figures are combined with features of non-flattering animals. The surprise here arises from the juxtaposition itself that, although improbable, carries some associative insight.

To distinguish between combinational creativity and the next two kinds, Boden introduces what is arguably the most important construct in her framework: conceptual spaces. These are structured styles of thought or ways of thinking. For example, ways of writing poetry, styles of sculpture, painting, or music, and even theoretical frameworks in disciplines like chemistry or biology. They are typically acquired from one's culture or peers, having a proven track record and deeply influencing what thoughts are possible - or impossible - to conceive. Whereas combinational creativity measures surprise by what is probable, the next two kinds measure it by what is possible. As Boden says, that "the world has turned out differently not just from the way we thought it would, but even from the way we thought it could."

Exploratory creativity operates within an established conceptual space, exploring at its boundaries to uncover new possibilities that were previously unnoticed. Transformative creativity is the most radical form. It does not merely explore boundaries - it fundamentally transforms them. This type of creativity challenges foundational constraints that define a conceptual space. In doing so, it creates the deepest form of surprise by making previously impossible ideas accessible.

But why constrain ourselves with conceptual spaces? Isn’t constraint the antithesis of creativity? Paradoxically, constraints unlock the deeper forms of creativity. As Boden notes, "to drop all current constraints and refrain from providing new ones is to invite not creativity, but confusion." We might not recognize it at first, as we constantly operate within the constraints of familiar conceptual spaces, but without these constraints, the number of possibilities to consider becomes literally astronomical. Recall the hypothetical Library of Babel, which contains every conceivable combination of letters and punctuation - encompassing all texts that have ever been, or could ever be, written. The scale of it is astronomical, though most entries are irrelevant. The value of a conceptual space and its constraints is that they filter out vast amounts of these irrelevant possibilities, increasing the likelihood of discovering relevant ones. Limiting possibilities thus allows us to explore more deeply.

Too abstract, give an example already!

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Having reviewed the key concepts, we can now better appreciate Kekulé’s iconic scientific insight into the benzene ring structure in 1865.

The conceptual space in chemistry from which Kekule was working was the current orthodoxy: that all organic molecules are based on strings of carbon atoms. In that conceptual space, the organic chemist’s job was to experimentally determine the elements and their proportions in a compound, then propose a carbon string structure consistent with both chemical bonding rules and observed laboratory behavior.

This methodology was successful for alcohol, and for many organic compounds like it. For instance, Kekulé himself published the representation of alcohol (CH3CH2OH) in 1861, as such a string of atoms, where carbon atoms size represent their valence, carbon being shown as 4 times bigger than hydrogen:

With benzene, however, this approach failed; the contradictions implied that no suitable structure could be found. Experiments showed benzene contained six carbon and six hydrogen atoms, yet a chain of six carbon atoms should have fourteen hydrogens, not six. The problem wasn't solved by suggesting double or triple bonds between benzene’s carbon atoms, since benzene’s chemical properties didn't support this. Extra bonds would have allowed benzene to readily bond with monovalent atoms.

After struggling with this issue for months, Kekulé had his famous insight:

"I turned my chair to the fire and dozed. Again the atoms were gambolling before my eyes. This time the smaller groups kept modestly in the background. My mental eye, rendered more acute by repeated visions of this kind, could now 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."

At this point in the book, Boden provides an interesting analysis of Kekule's daydream. The phrase “The smaller groups kept modestly in the background” indicates that Kekulé shifted his attention away from the problematic hydrogen atoms and toward the carbon structure. “Long rows” represent Kekulé’s adherence to the prevailing string theory. “One of the snakes had seized hold of its own tail,” forming the image of a ring. But surely rings and circles were common in Kekule’s environment, so why didn’t the insight occur sooner? The symbolism is deeper than it appears: a snake biting its tail represents a change in topology, from a typically string-like structure to an uncommon ring-like structure.

This analogy led Kekulé to a ring structure. However, forming a simple carbon ring with one hydrogen per carbon would leave each carbon with unused valency. Thus Kekulé proposed a further modification: alternating single and double bonds between the carbon atoms. Notice how both of these are relaxations of the constraints of the previous conceptual space, forming a new one - a clear example of transformative creativity.

Arriving at the structure above, there is still one question left: with all carbon atoms equivalent, how could the molecule "choose" which bonds were double or single? Kekulé answered that it couldn't — he proposed instead that single and double bonds spontaneously oscillate.

Due to these elaborate modifications, several prominent chemists dismissed Kekulé’s benzene theory as "a tissue of fancies" - a fragile combination of fancy ideas. But eventually the theory was vindicated. By shifting from string to ring, Kekulé opened a whole new science of aromatic chemistry and enabled exploring ring molecules with varying numbers and types of atoms. This is a typical characteristic of transformative creativity - what is found is not only a new creative possibility (benzene structure), but a whole new way of thinking (aromatic compounds).

But the three kinds of creativity are seen as a continuum. And even transformative creativity has its gradations. Take this progression.

Kekulé's insight of rings built from strings was undoubtedly creative. But conceptualizing string-molecules within atomic theory was even more creative. However Dalton’s transition from medieval elements (fire, air, earth, water) to modern elemental atoms was radically creative. Accordingly, Dalton holds greater historical significance in chemistry than Kekulé.

But the three kinds of creativity are in a continuum and even transformative creativity has its gradations. Take this progression. Kekulé's insight of rings built from strings was undoubtedly creative. But conceptualizing string-molecules within atomic theory was even more creative. However Dalton’s transition from medieval elements (fire, air, earth, water) to modern elemental atoms was radically creative. Accordingly, Dalton holds greater historical significance in chemistry than Kekulé.

What about exploratory creativity? Remember the work on alcohol, which was found by exploring the possibilities within the established conceptual space of the time. In fact, the distinction we are making between exploratory and transformative creativity has a strong parallel to the famous distinction of Normal and Revolutionary Science by Kuhn. In Normal Science, a widely accepted paradigm (think conceptual space) guides scientific practice and sets the stage for problem-solving within established rules. As unexplained anomalies start accumulating, scientists start questioning core assumptions, eventually shifting toward a new paradigm. That is Revolutionary Science.

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What about LLMs?

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A nineteenth-century chemist’s dreaming snake and a 21st-century language model’s fluent pastiche are both creative, but they are not equivalent. What kind of creativity, then, do large language models (LLMs) exhibit? When prompted to contribute in a brainstorming session, large language models can often be inspiring. Their responses are not merely copies of their training data; they exhibit creativity, at the very least, in the combinational sense of creativity.

Now, for exploratory creativity, we have seen some promising evidence, especially in large reasoning models or LLM systems that are placed in a loop of reflection. For example, in the literature on automating research via LLMs, there have been cases of LLMs generating surprisingly good paper ideas — ideas that a PhD student would be proud to publish.

But as Boden suggests, to explore deeply, we must inhabit constraints even if temporarily - we must have a conceptual space. But then, what can be said about the conceptual space of the AI system that came up with the paper idea? In theory, it is hard to even conceptualize because these models were trained on the collective externalized thoughts of all humanity. In practice, they are prompted to become specialists, for instance, to assume the role of a PhD researcher. Since LLMs can compartmentalize their abilities very well, such prompting might evoke a more coherent conceptual space - though it's still far from the real thing.

For better exploratory creativity, the models need to recognize the possibilities within the conceptual space they inhabit, that in turn will make their search more effective. At this time, it is not clear how far the chain-of-thought reasoning-style can take us in the creativity continuum. But to match the most creative between us humans and achieve transformative creativity, they will need to deeply understand the very constraints that underlie their thoughts and modify those constraints. In this case, it won’t be hard to tell when they do - the consequences in our world will be unmistakable.