Metaphors and Eureka moments

Maria Popova wrote a delightful blog post on the topic of creativity, in particular, the creative genius of the physicist Freeman Dyson. In it, she lets Dyson guide us through some of the events that culminated in perhaps his most celebrated creative breakthrough: the reconciliation of Feynman’s and Schwinger’s equivalent but distinct formulations of quantum electrodynamics (QED). Aesthetically, both approaches were ostensibly different. Feynman’s methods were highly graphic; teeming with iconic loops and squiggly lines that later would be simply known as “Feynmann diagrams” and reproduced by thousands of physicists. On the other hand, Schwinger’s approach was thoroughly mathematical. The equivalency between both methods at interpreting experimental data was by no means obvious. So much so, that Feynman and Schwinger saw their formulations as competing rather than complementary. Dyson became familiar with both frameworks as well as with both of their creators with whom he spent time discussing and studying. According to his account, the realization that both descriptions of QED were indeed equivalent came to him as a “flash of illumination” while ruminating on both frameworks during a bus ride back to his home in Cornell University. This stroke of genius that culminated in the unification of Feynman’s and Schwinger’s schemes earned Dyson a secure place in the history of physics. And that’s the story of these momentous events in a tiny tiny nutshell.

The “flash of inspiration” narrative is by no means rare among the stories recounting the moment of creative breakthrough.

In his book “The emperor’s new mind“, the physicist Roger Penrose includes Henri Poincaré’s personal account of the moment when his search for what he called “Fuchsian functions” came to a sudden end during a geological expedition:

…The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go to some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry.

After reading Poincare’s and Dyson’s stories one feels like cruising the roads in a bus stopping only to write down the next big breakthrough at leisure.

Anecdotal stories about what lies at the very onset of creativity can get more mysterious than that. Forget flashes of inspiration or raw divine intervention, sometimes people report that ideas “come down” to them in the form of, or thanks to, dreams. Surely the most famous of anecdote in this category is that told by the German physicist August Kekulé. Kekulé claims to have envisioned the hexagonal structure of the benzene molecule in a dream. In the dream, dancing atoms formed a closed string that subsequently trans-mutated into a snake eating its own tail. Here it might be prudent to point out that this fantastic story has its detractors.

In any case, what’s the deal then? Is inspiration handed down to us in the form of dreams or visions coming straight from some mysterious divine source? Are the doors of buses inter-dimensional portals to “Eurekaland”? Just how mysterious is the mystery behind the machinery of creative thought?

Penrose himself starts peeling off the veil of mystery when he writes:

… the question of what constitutes originality should be raised. It seems to me that there are two factors involved, namely a “putting-up” and a “shooting-down” process. I imagine the putting-up could be largely unconscious and the shooting-down largely conscious. Without an effective putting-up process, one would have no new ideas at all. But, just by itself, this procedure would have little value. One needs an effective procedure for forming judgements, so that only those ideas with a reasonable chance to success will survive. In dreams, for example, unusual ideas may easily come to mind, but only very rarely do they survive the critical judgements of the wakeful consciousness… … In my opinion, it is the shooting-down (judgement) process that is central to the issue of originality, rather than the unconscious putting-up process; but I am aware that many others might hold to a contrary view.

That is as good a beginning as anything. Penrose’s view is mechanistic, in contrast to the quasi-mystical “flash of inspiration” narrative. Although, maybe not mechanistic enough. Magic could be hiding at the very onset of the “putting-up” process. Its unconscious character shields it from analytic conscious scrutiny pretty much by definition. We (and many others) argue that, unfortunately for the mysticism enthusiasts, Penrose’s putting-up process is not fueled by magical forces or divine intervention.

Tapping into the creative force of metaphors.

Pierluigi Assogna opens his short paper “Similes and metaphors for creativity” with a concise statement:

“The importance of metaphorical thinking in promoting creative ideas has been widely discussed for many years…”

Where there is so much smoke one reasonably intuits the presence of a fire. We could grant that indeed it might be the case that metaphorical thinking promotes the sprouting of creative ideas. But then, what exactly is it about metaphors that could give sustenance to that claim? That is a question worth exploring.

A question like this could accept various answers depending on the level of analysis that we want or can probe. The ultimate bottom-up type of answer would have to start by trying to establish the neurological basis of metaphorical thought and build up from there. It might necessitate to put experimental subjects into a MRI scanner, set up a situation where they have to “get creative”, scan their brains while they perform whatever task has been assigned to them, see which parts of their brains “light up” and in which sequence, compare the patterns of activity to those previously obtained from people engaging in constructing metaphors and look for correlations. This is what we will not do because there are scores of reasons that absolutely disqualify us from taking such an approach. Besides, there are people trained in everything one needs to be trained in order to study the neurological underpinnings of creativity and, luckily for everyone else, they have written extensively and in depth about it. Also there are those who have summarized the findings and presented them in an accessible way for popular consumption. Instead, we will attempt to present what could be called a “cognitive case” for the link between metaphors and creativity that falls somewhere in between the hard, rigorous neurological approach and purely subjective experiential accounts.

Metaphors as tools for cognition.

Perhaps the most familiar definition of metaphor is that a metaphor is essentially a figure of speech that uses one thing to mean another making a comparison between the two. However, this definition does not sufficiently expose the most interesting and indeed defining characteristic of a metaphor, that is, its conceptual aspect.

The two most responsible persons for launching a cascade of research into the cognitive character of metaphors are undoubtedly George Lakoff and Mark Johnson. They define a metaphor as:

a cognitive process that allows one domain of experience, the target domain, to be reasoned about in terms of another, the source domain. (our emphasis).

From this perspective, the “figure of speech” definition only accounts for a single manifestation of metaphorical thought.

There is a rule of thumb that most metaphors follow: the target domain tends to be more abstract than the source domain, which conversely, is rather concrete. It is in this abstract-to-concrete mapping where the clarification power of metaphors lies.

It is then fitting to use a concrete example to highlight how metaphors bring home hard-to-grasp concepts by linking them to more concrete ideas. One of the metaphors we live by is the love is war metaphor. It becomes manifest every time we say or hear things like: “we need to fight for this relationship”, “You need a strategy to conquer her heart”, etc. We borrow concepts from the source domain war to explain features of the more abstract domain love.

But not every instance of metaphorical thought applied to speech is that conspicuous. Metaphorical speech is in fact ubiquitous but it is generally so subtle that it routinely escapes our notice. When we say “that price is too high” we are applying the up is more metaphor. When we say “I should spend more time with my kids” we use the time is money metaphor. When we say “it is good to plan ahead” we display the front is future metaphor. And so on and so forth. In closer inspection, it seems that we cannot talk without applying metaphors. Metaphors pervade language. Every language. Metaphors help to generate language more than the other way around. Essential aspects of thinking itself are metaphorical in nature (the core of Lakoff and Johnson’s thesis). Harvard psychologist Steven Pinker touches on this idea by stating:

Mental metaphors are innate. Cross-domain mappings are the result of co-opting neural machinery that evolved for perception and action to support more abstract thinking.

The “cross-domain mappings” that are the key operations of metaphors are the substrate upon which a lot of our cognition is built. More than that, no one taught us to think this way. Evolution built into us algorithms to conceptualize the world that use metaphorical mappings:

  1. We perceive the world.
  2. We break it into pieces (objects and actions) that we group into conceptual domains based on how often the pieces occur together or cause each other.
  3. We bridge the domains through metaphorical-style thinking and that endows us with the capacity to abstract principles and relations to better understand the world.
  4. Repeat the cycle to refine understanding.

This view of metaphors puts conceptual mapping at the center of our cognition. One can argue that this type of cognitive capability engenders or at least potentiates our capacity to reason; an ability from where things like science and philosophy stem. We started with a humble definition of metaphor as a mere “figure of speech” and ended up discovering that this non-assuming concept might be at the core of what makes us distinctively human.

Analogies = Metaphors unpacked.

Take now the pervasive metaphor “ideas are food” that comes forth when we say things like:

  1. Yesterday’s lecture was pure food for thought.
  2. I am still digesting that last thought.
  3. I will chew on this idea and get back to you.
  4. What you just said left me with a bitter after-taste.

Sentence 1 pretty much defines the metaphor. The subsequent sentences allow us to get a better intuitive grasp on various complicated cognitive processes involved in handling ideas. Sentence 2 expresses how analyzing and understanding an idea is a lengthy and energy-consuming process just like digestion. Sentence 3 suggests that examining an idea in real time is akin to actively and arduously chewing on something before we internalize it, before we “swallow” it. Finally, sentence 4 reminds us of how after we have “chewed” on an idea and fully “digested” it, its implications can instigate a value judgment in us (a negative one in this case). Pursuing a metaphor progressively deepens our understanding of a complicated concept by looking at the various ways in which it can be linked to a simpler one.

Isolated and scattered metaphors allow us to inspect an abstract concept from one angle at a time. Mixing various metaphors in a single “train of thought” highlights more aspects of the concept we want to explain in a sequential order. However, one needs to jump from one metaphorical framework to another. This muddles the full clarifying power that is attainable by sticking to a single metaphorical theme. Take a look at this particular example designed to illustrate the point:

Nowadays it is very fashionable to think in terms of networks. Antiquated conceptions of isolated agents won’t cut through the challenges of today. These type of ideas simply ran out of power.

Three metaphors have been sequentially displayed by our hypothetical speaker: ideas are fashion (fashionable, antiquated), ideas are sharp objects (won’t cut) and ideas are machines (ran out of power). Together they get the message across, the speaker is urging us to abandon the reductionist “isolated agent” paradigm to adopt a more holistic “network paradigm”. Metaphors deployed in this way are good enough for many purposes of communication… it is just that metaphors can do much more when pursued more systematically in order to extract the most out of them. That is what analogies do. Analogies use metaphors to justify their own aptness.

When you offer an analogy, essentially what you are doing is taking a listener through a systematic field trip through a well-defined metaphorical map. As we have seen, metaphors build multiple bridges between two conceptual domains; an analogy explores several of those bridges while simultaneously explaining the reasons why they are sound. Let us use an example to illustrate this. Take the metaphor “the brain is a computer”. Now, let’s make an analogy out of it:

The brain is like a computer insofar as it is an information processing device. It returns an output to multiple types of stimuli that constitute input signals. When you lock your attention on a task, you make use of as much working memory as the task demands. Whenever information that is not readily available is needed, you browse through your stored “read only memory” database to retrieve the necessary data. The information itself is coded in complicated circuits made by neurons. At first approximation, the neurons are binary systems with on and off states: firing and non-firing. If useful data is not found, then you need to compute an entirely novel solution.

All the bold words reveal the underlying metaphor. The full analogy unpacks the metaphor and turns into a didactic device.

Metaphors are ubiquitous and come naturally to us all, we can’t communicate efficiently without them. In contrast, analogies are arduous and deliberately crafted. They are like rosetta stones that teach us how to systematically translate elements from a conceptual domain into another. But there is more. Given that to construct an analogy you need to switch yourself into fully introspective analytical mode, analogy-crafting often illuminates new ground that had remained concealed from your own mind’s eye. A similar phenomenon is experienced when one attempts to articulate complicated ideas in writing. Writing is like steroids for the thought.

If analogies have the power to focus your attention onto information that you already possessed but never seriously engaged with or even consciously noticed, then, they should be seriously considered as tools to pump up the creative muscles of your brain.

The metaphorical nature of creativity.

By now it should not seem so arbitrary to speculate that something like metaphorical thought could be near the center of creative thinking. Specifically, the task of generating connections between two dissimilar conceptual domains and then consciously tie them together in an analogy-type scheme. Maybe a process of this sort, at least partially, demystifies the quasi-magical subjective feeling of the “Eureka moment”. Possibly scattershot metaphorical inter-domain bridge building is at the core of Penrose’s initial “putting-up” stage of the creative process and maybe something like the coherent integration of metaphorical links into an analogy-like structure is the culmination of his “shooting-down” stage. Maybe.

But then, keeping this framework in mind, what do we make of, for example, Einstein’s description of how he perceived the quality of his own creative thought process to be?:

The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements of thought are certain signs and more or less clear images which can be ‘voluntarily’ reproduced and combined… The above mentioned elements are, in my case, of visual and some muscular type. Conventional words or other signs have to be sought for laboriously only in a second stage, when the mentioned associative play is sufficiently established and can be reproduced at will.

Does a dynamic like this one fit with what we have talked about so far? Maybe partially. Let’s start at the end. Einstein considered that verbalizing the scene played out by the visual elements in his imagination using conventional language was the final and ‘laborious’ stage of his thought process. This deliberate, formal, difficult and final verbal structuration strikes us as very similar to the analogy-building concluding step that we placed at the end of the creative thought process.

The hurdle separating his account from conforming to our toy model rooted in metaphor seems to be found in the unfolding of his language-free creative process. He claims that his “mechanism of thought” is first and foremost geometrical.

The first thing to do would be to recognize that language-free thought is absolutely possible. Luckily, the typical objections to the possibility of thinking without language can be quickly dismantled. Here is a simple thought experiment that demonstrates that.


In a lengthy video titled “linguistics as a window to understand the brain” Steven Pinker asks us to look at the two constructions shown above. He then asks if they have the same shape? The answer is “yes”. The way most of us reach that conclusion is by mentally rotating one until it matches the other. What we do not do is to narrate to ourselves the progress of the mental rotation as we perform it.

Was any metaphorical thinking needed to perform the mental rotation? Not really. Is there any fundamentally metaphorical operation needed to play out mental movies whose protagonists are elements of various shapes, or psychical entities as Einstein calls them? Not really. To be clear, not every form of thought is metaphorical in nature. That is not the claim.

Language-free geometrical thinking is not necessarily metaphorical. Although, it can promptly acquire metaphorical characteristics when it is used to visualize a real world phenomenon, which is very likely the kind of thing that Einstein was doing when musing over physical problems.

How could the psychical elements of Einstein’s mental movies have looked like? We cannot know for sure but a proverbial “educated guess” is warranted by looking closer to his scientific writings. In his landmark paper “Concerning an Heuristic Point of View Toward the Emission and Transformation of Light” which is usually known as his paper on the photoelectric effect (for which he got a Nobel Prize) Einstein wrote:

…the energy of a light ray spreading out from a point source is not continuously distributed over an increasing space but consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed as complete units.

This passage opens a door to peer into his imagination and speculate about details of the psychical actors playing the role physical elements in his mental movies. The light quanta which are localized at points in space that move as whole elements without dividing description conforms neatly to the “light is a particle” concept which has a quintessential metaphorical structure. Now, to say that “light is a particle” or its classical counterpart “light is a wave” are metaphorical modes of thought is not to say that they are only that. It is up to experiment to decide whether light is indeed a particle or a wave in which case either metaphor would instantaneously transform into a factual description of light. However, as nature would have it, light has a thoroughly schizophrenic character sometimes behaving as a particle (as in Einstein’s analysis of the photoelectric effect) and sometimes behaving as a wave (as in the interference patterns formed by electrons in the “double slit experiment“). Today the “particles of light” or photons are better conceived as localized vibrations of the electromagnetic field. A quite more abstract and refined notion than either photons being particles or waves.

What concerns us here is that sometimes light is imagined as an actual ball-like particle and sometimes as an actual wavy-wave. And that both conceptualizations explain some of the properties of real light. It is not a huge stretch to imagine that the character that the “mental figure” playing the role of light in Einstein’s movie of the photoelectric effect took a particle-like guise. This is essentially a metaphorical mode of thinking, mentally replacing the more abstract concept light for the concrete concept particle and then letting the movie roll.

If you imagine photons as particles they are likely to behave in a way that is consistent with that conceptualization in your thought experiments. If you picture them as waves, then they will act differently. Both metaphors have explanatory power. You need to pick the metaphorical representation that better suits your thought experiment. You cannot have it both ways.

Metaphors and scientific breakthroughs.

The conceptualization of light as a wave got a strong footing thanks to Maxwell’s synthesis. Then Einstein showed that light could also behave as a stream of particles or photons. Electrons where first thought of as particles. Later on the aristocrat and physicist Louis Victor de Broglie showed that a wave-like description of electrons is also consistent. These conceptualizations are much more than just metaphorical flights of fancy. As we hinted at earlier, they had a shot at being true-to-the-bone objective descriptions of real light and/or real electrons. But as they stand now, both are representations that are so internalized in the minds of physicists that their metaphorical character is hardly ever recognized by them.

Other metaphorical more explicit allusions can be easily found engraved in the very names of various scientific ideas. We have black holes, the holographic principle, the Big Bang. These titles instantaneously evoke concrete concepts (holes, holograms, explosions). Moreover, when we verbally explain what any of these physical concepts we inevitably employ metaphors and analogies to some degree. The analogies address various questions: In which sense is an astronomical black hole like just a black hole? To which extent one can understand the Holographic principle by understanding how a hologram works? How much was the initial expansion of space and time just like an explosion? etc.

From the perspective of this post a different kind of questions is more interesting: How did thinking about concrete holes, holograms and explosions helped scientists to better understand black holes, holographic principles and Big Bangs? Did these concrete concepts play a pivotal role in the mental movies running in the minds of the scientists that conceived the theories? To which extent fleshing out the analogies and pinpointing the points of departure helped in refining the theories? Where did the power of imagination prove insufficient and the mathematics had to take the lead? There is likely no generic reply to any of these questions and the answers surely vary widely from one case to another. The Holographic principle is notably less intuitive than the Big Bang, surely the unaided mind is able to make more progress in the latter than in the former.

However, there are numerous cases in which scientists explicitly urge us to apply metaphorical transformations that have good chances of producing scientific progress. They ask us to transplant a concept from one domain into a second domain where they expect it to play a clarifying role. For example, with some dismay, Geoffrey West writes in his book “Scale“:

Energy is primary. It underlies everything that we do and everything that happens around us … This may seem self-evident, but it is surprising how small a role, if any, the generalized concept of energy plays in the conceptual thinking of economists and social scientists.

He indirectly urges economists and sociologists to import the concept of energy from physics and include it in their working tool kits.

A little earlier in the book he points out at a metaphorical transaction centered around the concept of metabolism which is already reaping some fruits. Talking about natural and social systems like organisms, cities and companies, West writes:

None of these systems, whether “natural” or man-made, can operate without a continuous supply of energy and resources that have to be transformed into something “useful”. Appropriating the concept from biology, I shall refer to all such processes of energy transformation as metabolism.

The metaphor is not just an inert, if clever, figure of speech, it is fully functional!

One can find many more examples of truly awe inspiring creative breakthroughs rooted in metaphorical thinking. So many that there is no space to comprehensively sift through them here. We will just shout out to one of them because it was mind-blowing to us when we encountered it and then another because of its monumental relevance.

The first is the incredibly unlikely connection drawn by Ginestra Bianconi between a model for the World Wide Web as described by network theory and… the Bose-Einstein condensate! But, if you want to learn more about it, well, actually anything about it, you can dive into the paper here or get yourself a copy of the great book “Linked” by Albert-László Barabási (her scientific advisor) and check it out there. You won’t be disappointed.

And, faithful to the tradition of saving the best for last, we need to remember the metaphorical nature of perhaps the best single idea anyone has ever had: evolution by natural selection.

Darwin dedicates a large initial chunk of the “Origin of species” to talk about breeders. Pigeon breeders most of all. He meticulously explains how the physical characteristics of pigeons can be molded by the artificial selection of human breeders to the extent of creating barely functional creatures like tumbler and fantail pigeons or nasty looking monstrosities like dragon pigeons. (in all fairness, he did not describe them in such contemptuous terms). Only after the workings of artificial selection are well established he made the move of replacing “human action” as the selective agent and putting the natural environment in its place. By performing the quintessentially metaphorical operation “nature is a choosy agent” he got us the theory of natural selection. Of course there is much more to the story (do the Beagle and the Galapagos islands ring a bell?) but it is clear that thinking about nature as an agent gradually modifying organisms through countless generations played a big role in his thought process.

So, are metaphors just handy figures of speech? Think twice.

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