Friday, October 25, 2013

Metaphors and analogies: scientists’ friends or foes?

Metaphors are like men. If you take them
seriously the joke's on you.
The girls were about a year old and I was working from home. As I was reading yet another referee report that came with the preamble “we regret to inform you...,” I watched Gloria trying to push a square block through a round hole. We were really trying the same thing, I thought.

Decoding metaphors and using analogies is a prototypical right-brain task, a pattern finding that helps us get a grip on new situations quickly and that sheds new light on the familiar. Metaphors and analogies are omnipresent in literature and the arts, in humor and also in education. And popular science writing is full of it.

But relying on metaphors is like traveling to a new country and then heading to Starbucks. The very reason to do it is also what limits the experience. It’s familiar and easy to understand, but it prevents us from learning something new. This is why I have a love-hate relationship with Starbucks and other metaphors.

Love: Analogies and metaphors build on existing knowledge and thus help us to understand something quickly and intuitively.

Hate: This intuition is eventually always misleading. If a metaphor were exact, it wouldn’t be a metaphor.

And while in writing, art, and humor most of us are easily able to tell when an analogy ceases to work, in science it isn’t always so obvious.

When it comes to physics I can most often tell when an analogy fails to capture the actual science. But in other areas of science this sometimes is not clear to me. There are for example these artistic images that frequently accompany popular science accounts of new drugs or cancer treatments. You know, the ones with the molecules that fit like keys into locks of other molecules, or that cut through molecular bonds. I am reasonably sure that these explanations suggest a clarity of the underlying mechanism and structure that most often doesn’t quite exist in the actual data. But how much of it is science and how much of it is art is difficult for me to tell.

In a recent Nature comment “Mind the metaphor”, Eleonore Pauwels made a similar point:
“[I]n the late 1990s, computer scientists, physicists and engineers were fuelled by the idea that they might be able to direct cells in the same way that people program computers. In the laboratory, researchers started to use computing and engineering metaphors –switches, oscillators and logic gates, for instance – both to guide the design of synthetic constructs and to understand how natural systems function. Almost immediately, scientists were confronted with the uncertainties and constraints of engineering in the cellular context. Engineering concepts and metaphors could serve only as an inspiration...

Scientists using metaphors among themselves are often aware of, and even careful to point out, the subtleties that could be misconstrued. Problems tend to arise when metaphors are used outside the laboratory...

Faced with explaining the messy complexity and uncertainty of science to the public, it is understandable that scientists reach for metaphors. But [this] sends a message to policy-makers and laypeople that scientists can already make biological systems that are reliable and controllable. It widens rather than closes the gap between scientific realities and the expectations of policy-makers and the public.”
The same problem exists in physics, though at least in the area I work in there aren’t all that many implications for public policy. But I’ve seen it over and over again that people take analogies too seriously and start trying to build arguments on them. Suddenly a rubber sheet isn’t just an analogy for space-time, but it is space-time. The universe is an inflating balloon, the Higgs particle is a rumor, and entangled particles are shoes in parcels.

Except that, well, they’re not. The universe isn’t a clockwork and it’s not a drum either; the brain isn’t a computer, black holes are not cannibals and indeed not even black.

The main reason we use mathematics for scientific theories is that it’s a particularly clean way of thinking, uncluttered from what the right brain wants to associate. An electron isn’t a spinning top, it’s an element of a Hilbert space that transforms under the spinor representation of the Lorentz-group. There is really no metaphor that’ll do equally well. Feynman diagrams seem to be particularly prone to misinterpretation as many people believe they depict physical particles, while they are actually a handy short-notation for lengthy integrals.

But my uneasiness with metaphors and imagery goes beyond the communication issue.

If you spend some time with a set of equations, pushing them back and forth, you’ll come to understand how the mathematical relationships play together. But they’re not like anything. They are what they are and have to be understood on their own terms*.

Thus, as much as I value metaphors for the intuition that can serve as a guide to new ideas, I also mistrust them. We learn much more from the failure of metaphors than from their success.

I admired Gloria for her persistence in trying to push the square block through the round hole. Then Lara took the piece out of Gloria’s hand, opened the lid of the bucket and put the block in. Problem solved. If only it were so easy with my papers…

*That is unless you are onto a theory that is truly equivalent (‘dual’) to some other theory.


  1. See, that metaphors are so central to our thinking is exactly what worries me. I am wondering if not this ultimately limits what we can understand. Quantum foundations are a case in point. How often do people try classical analogies to explain something which really just can't be explained this way? Are we doing anybody a favor with this? There isn't really any substitute for equations I think.

  2. "fit like keys into holes"

    The English phrase is "keys into keyholes" or perhaps "keys into locks". (It just occurred to me that lock, at least in this context, might be related to the German "Loch", meaning, of course, "hole", but there doesn't seem to be any close relation.)

  3. Crystal structures' PowerPoint talks can miss active sites being dynamically clever. Small molecule lock-and-key therapeutics are now astoundingly expensive protein mimics. Human Resources' well-dressed lobotomites to the halcyon epheremedes of MBA omniscience, Big Pharma's ventures are now dry wells. What killed innovation? Metaphoric coarse grids.

    Rubber sheet gravitation is hyperbolic geometry. A gravitational well is elliptic geometry. What could go metaphorically wrong, education to application?

    The big fun hides in small footnotes, the working ends of ice picks. Metaphors scrub surfaces facile but coarse-grained smooth. Organic chemistry is done jury-rigged LCAO not computed MO. The divergence is the Woodward-Hoffmann rules, among other things. Good enough for application is often not good enough for discovery. Leap! A net will appear beneath you more often than during planned descents into local minima.

  4. Once we are convinced of the reality of something, the mathematics, too, is only a metaphor for that thing.

    "An electron isn’t a spinning top, it’s an element of a Hilbert space that transforms under the spinor representation of the Lorentz-group."

    Tomorrow the electron may be a different mathematical object which looks like an element of a Hilbert space that transforms under the spinor representation of the Lorentz group in suitable limits.

    It is metaphors all the way down.

  5. Sabine,
    Metaphors aren't so much about building accurate models but rather they are used to decide which road to take when there are many roads available and the empirical evidence is not yet clear to decide which one.

    In other words, metaphors are valuable in deciding where to devote your energy, everything else being equal. That is THE most important use of metaphor.

  6. Two nice podcasts on (ab-)using metaphors. By a linguïst:

  7. "And I cherish more than anything else the analogies, my most trustworthy masters. They know all the secrets of Nature, and ought least to be neglected in Geometry."
    - Johannes Kepler

  8. Hi Bee,

    Analog models of quantum field theory in curved space

    In condensed matter, one can construct systems where the propagation of long wavelength phonons (sound waves) is very similar to the propagation of a scalar field in a curved Lorentzian spacetime. Such systems are called 'analog models'. It is even possible to construct analogies to black holes in this manner, where the phonons that travel past a certain point cannot return. For example, consider a fluid where long wavelength phonons in the fluid propagate with speed cs, which is analogous to the speed of light in these models. Now put this fluid in a pipe and change the shape of the pipe such that the speed v of the fluid is faster than cs in one section and slower in an adjacent section. A phonon can travel "back against the current" only up to a certain point, where the the fluid speed equals cs. After that the fluid flow carries it down the pipe. This point in the pipe therefore mimics a black hole event horizon, from which nothing can escape. Other black hole features such as Hawking radiation are also present in these models. Since these models give an example of a system that has a fundamental structure at very short distances (where the fluid description breaks down), yet has a pseudo-Lorentz invariance at long distances.
    David Mattingly

    You see how when you "set the stage" how important cross fertilization is and how important cross fertilization is used by individuals in different careers to bring other areas of research new perspectives.

    Most famously Stuart Kauffman, and others who see the need to have theories extended to be able to validate the foundation of biology most appropriately. Arun used mathematics, and most respectively, addresses your point about delving into the simplicity of the correlating subject by using only mathematics. Where have we heard about such abstractions before? :)

    "Fluid dynamics offers a mathematical structure, which underlies these practical discipines, that embraces empirical and semi-empirical laws, derived from flow measurement, used to solve practical problems. The solution of a fluid dynamics problem typically involves calculating for various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time"


  9. The Navier–Stokes equations dictate not position but rather velocity. A solution of the Navier–Stokes equations is called a velocity field or flow field, which is a description of the velocity of the fluid at a given point in space and time. Once the velocity field is solved for, other quantities of interest (such as flow rate or drag force) may be found. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid; however for visualization purposes one can compute various trajectories.

    As you delve deeper into the presentation of astrophysical article, you realize how much more must be added, to understand how perspective from Crab Nebula, is understood.


  10. Bee,

    It seems that the capacity to determine that one thing is like another must be a foundational attribute of living systems. Perhaps one could trace its fundamental origins back to simple chemical, boundary interactions of the "lock and key" sort you mention. The ant finds its path marked by invisible cairns.

    We are wired to make metaphor. As social beings with a motor cortex predominately devoted to running our mouths, the chemistry of association has grown wondrously complex and multi-dimensional. All our senses are involved. We take delight in the new associations created by metaphor, but more than that, they often guide us.

    “Nights candles are burnt out, and jocund day stands tip toe on the misty mountain tops,” and you worry that reliance on metaphor will lead us to an unbridgeable gulf, that the “this-is-like-that” of our work-a-day world will limit our understanding of some deeper quantum reality, that only mathematics can span the gap.

    Yet, as has been noted, mathematics is an explicit form of metaphor. I worry that for me, practically speaking, there is an unbridgeable gulf between that realm of metaphor and myself. I also worry that mathematics may be so compelling for some that it mistakenly projects its own reality upon nature.

    So, a tear wells up in my eye “like a fat lady leaning out of a tenement window.”

  11. If the mathematics used in physics is not a metaphor for reality, then we must admit we live in Tegmark's universe.

  12. Imagine, a Quantum Biology.

    Does this actually bring you into line with what we know about the quantum reality and what nature has to offer of herself? You see then how quantum mechanics has been used as a metaphor for how we can describe nature.

    So for however abstract with regard to the simplicity and beauty that such math may reveal, there is a basis in the natural processes?


  13. Well, there are good metaphors - an analogies and bad metaphors - a homologies. The blind usage of metaphors is as two-sided as the blind usage of formal math. Both they're just a tools for better understanding of reality - no less, no more. So, why not to use a metaphor for it?

    We could compare these tools to the sculptor tools used during formulation of ideas. The metaphors are rough tools, which are required for getting the shape at the beginning of sculpting. The tools like the math analysis is good for finishing - but they're pretty slow and tedious to use. In addition, when you work on tiny details, you can lost the meaning of the global shape easily. Your statue will be nicely polished, but still ugly shaped.

    Now, the problem is, that the contemporary physicists aren't payed for actual results, just for time. So they learned to use the finishing tools from beginning to end. And the effectiveness of development of their theories became suboptimal. They developed a bunch of ugly theories, which they don't actually understand so they cannot know, if they lead to the correct result. So that they just repeat: hell, give us a more time!

  14. In AWT the role of metaphors and deterministic thinking is analogous to spreading of longitudinal and transverse waves during information spreading. These two mechanisms tend to alternate during topological inversions of causal space-times in similar way, like the space-time at the event horizon of black hole or particle horizon of observable universe.

    That is to say, the formal models lead to exact but fragmented description of reality (landscapes of stringy and loopy theories as an example). When the number of ideas increases some level a new high level ideas will start to emerge from it on background of metaphores. The metaphores are decreasing entropy of our thinking - they're essentially telling us, that two or more seemingly unrelated concepts have something in common. They lead into unification of our theories.

    But the metaphors are inherently fuzzy, so that when the most successful metaphor will emerge, it will become concertized in formal models and it gradually becomes fragmented again - so that whole process will repeat in nested way in process of cyclical evolution. For example today the physicists are forced to return to idea of aether, which was popular in 19th century.

  15. /*Fluid dynamics offers a mathematical structure, which underlies these practical disciplines*/

    In dense aether model the metaphors repeat at various dimensional scales again and again. This is because with increasing amount of particles their system will become elastic, then fluid, then turbulent, then elastic foam, then fluid again, etc. So that the fluid metaphor repeats in dense aether model at various scales, but we can say the very same about Boltzmann gas or quantum foam metaphors.

    There is no more universal metaphor or phase of aether, than another ones. All they're emergent and transforming one into another in physical hyperspace. The usage of models and metaphors in human civilization just replicates this emergent paradigm in causual hyperspace.

  16. Hi,
    You have captured the essence of an issue that I have been frustrated with for some time. Not my biggest peave with poor science writing, but one of them.

    Metaphors are a tool used extremely frequently. More often than not, they fail. There are two failure modes: (1) The metaphor misses the central and most important feature or concept, or (2) The metaphor misleads by giving the reader the impression that some property of the metaphor applies to the physics concept under discussion.

    Like a scalpel in the hands of a skilled surgeon, if used correctly a metaphor can produce positive results; advance knowledge, comprehension, and interest. But, in the hands of someone who is not skilled or not careful, a metaphor can do more damage than good. Far too often, the writer fails to think about the different ways that the reader could interpret the metaphor.

    At the very least, metaphors need to come with warning labels. I recall from my own experience: I learned about General Relativity from popular science books and the metaphors and simplifications that they contain. When I took a formal GR course in college, I had to struggle to replace all of the mis-information that filled my head, that I had acquired from the popularization books.

    I could go on with other examples where metaphors in the classroom or in text books have done more to confuse me than to help me.

    Bottom line is, I think metaphors may have a place in popular media and science outreach WHEN USED BY A SKILLED AUTHOR. But, they should be avoided like a plaque in classrooms and physics textbooks.

  17. Dr. Bee,

    Perhaps there is the tantalizing possibility of “deep metaphor,” some commonplace phenomenon encountered in a morning walk that, if we only knew how to interpret it, would echo nature’s first fetal heartbeats.

    For all its apparent diversity the universe is also manifestly self-similar. The soliton’s essential dynamics were first observed from horseback. Perhaps, if we put ourselves in the right place, a Eureka moment will drop from an overhanging bough.

  18. Douglas Hofstadter (Godel, Escher, Bach) has a new book out with Emmanuel Sander, "Surfaces and Essences: Analogy as the Fuel and Fire of Thinking." As I usually do, I skipped to the end, Chapter 8, "Analogies that Shook the World."

    They describe in detail how they perceive that Einstein used analogical thinking to make many of his intuitive theoretical leaps. Of course, Einstein knew that the analogies were just that, but Hofstadter and Sander claim that he used them as a guide.

    Worthwhile reading

  19. Arun,

    Yes, I agree with you. But many science writers think maths is a 'metaphor' that is difficult to understand, so they're looking for a metaphor that is easier, which most often means it's more familiar or more intuitive. What I'm saying is that whichever way you put it, if it's not the mathematical model that according to present knowledge is the best we can do in describing nature, it is not only inaccurate but also eventually misleading and hindering a deeper understanding. Best,


  20. Hi Philip,

    Thanks, I fixed the sentence with the holes that are locks :) Best,


  21. Eric,

    "metaphors are valuable in deciding where to devote your energy, everything else being equal. That is THE most important use of metaphor."

    I don't share this opinion. The most important use of metaphors is in my opinion to convey technically difficult topics and to raise interest in understanding the technical details. That can only work however, if it is made clear in using the metaphor where its usefulness ends and how to move on to the more accurate description. Most science writing falls short of this.

    I would agree that metaphors are a useful source of creating ideas, but they're certainly not the only source and it's hard for me to tell just how relevant this particular source is. I know that a lot of my colleagues are intrigued by quite technical problems whose solution they try in the end to phrase in terms of simpler metaphors for the sake of communication, but the interpretation very often follows the maths and not the other way round. (In fact, it normally doesn't work the other way round, at least for me.) Best,


  22. Plato,

    In fact the analogue gravity has been on my mind when I wrote this article. I have been wondering for a while whether this is a pointless cosmetic exercise that just thrives on promoting fluid dynamics as 'sexy' because, eh, now it's got something to do with quantum gravity. Or if directly modelling one system with another system rather than modeling both by maths is an important change in the scientific methods that we use. Best,


  23. Warren,

    Indeed, I have made the same experience with popular science books I read as teenager whose misconceptions I had to clear out later. Now, needless to say, I am worried that I have similar misconceptions in other areas of science where I never read any textbooks. Best,


  24. Anthony,

    Yes, that's exactly what worries me you see. I have no doubt that much of scientific progress has come about by the use of analogies. But how far can we possibly go with this. And Einstein is a point in case, as he had evidently great difficulty accepting quantum theory for its lack of analogy. Best,


  25. "They describe in detail how they perceive that Einstein used analogical thinking to make many of his intuitive theoretical leaps. Of course, Einstein knew that the analogies were just that, but Hofstadter and Sander claim that he used them as a guide."

    Yes, Einstein wrote that he thought in images, not so much in mathematics. He was forever grateful to various mathematicians who helped him out, from Marcel Grossmann to Walther Mayer. He did realize, though, that it was essential to communicate his results in mathematical form.

  26. Eric and B.,
    I agree with Eric that metaphors are one tool that you can use “in deciding where to devote your energy,” but I also agree with B. that they are definitely not the only tool, and it probably is impossible to decide how valuable they are in general. Physicists (scientists in general) probably don’t even know themselves how much they use them. They may make up a story – after the fact – that supposedly describes how they came to some epiphany, but that’s hindsight.
    And B., it is true that analogy will only take you so far, but I disagree that the lack of analogy is the reason why he had difficulty accepting quantum theory. He used some level of analogy in developing his quantum theories – primarily with classical thermodynamics. However, his drive for a more complete theory than quantum mechanics came about after thinking deeply about the subject for many years. In fact, he – along with Bohr – was probably one of the people who had thought the most deeply about quantum theory, and came to his conclusion after looking at the subject in many ways – both analogically and mathematically.

  27. /* I have no doubt that much of scientific progress has come about by the use of analogies. But how far can we possibly go with this..*/

    Definitely not at the end, but the perspectives of imagination are definitely better than before one hundred years. Before one hundred of years the human understanding leaved the high dimensional scope of reality - now it returns into it again.

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  29. Anthony:They describe in detail how they perceive that Einstein used analogical thinking to make many of his intuitive theoretical leaps. Of course, Einstein knew that the analogies were just that, but Hofstadter and Sander claim that he used them as a guide.

    I believe there is something to this that is worth thinking about. I tried to track it down as to the times in Germany when visualization and geometries were a very important attribute sought toward developing models.

    I have mentioned quite often Dirac in this same way.

    PAUL DIRAC:When one is doing mathematical work, there are essentially two different ways of thinking about the subject: the algebraic way, and the geometric way. With the algebraic way, one is all the time writing down equations and following rules of deduction, and interpreting these equations to get more equations. With the geometric way, one is thinking in terms of pictures; pictures which one imagines in space in some way, and one just tries to get a feeling for the relationships between the quantities occurring in those pictures. Now, a good mathematician has to be a master of both ways of those ways of thinking, but even so, he will have a preference for one or the other; I don't think he can avoid it. In my own case, my own preference is especially for the geometrical way.

    Felix Klein on intuition

    It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction ....

    I am led to these remarks by the consciousness of growing danger in Germany of a separation between abstract mathematical science and its scientific and technical applications. Such separation can only be deplored, for it would necessarily be followed by shallowness on the side of the applied sciences, and by isolation on the part of pure mathematics ....

    Maybe Felix's statements speaks to the idea of the necessity of going slow with it and developing mathematically because soon it can become an abstract feature pushing forth through for further development.

    Intuition and Logic in Mathematics by Henri Poincaré

    On the other hand, look at Professor Klein: he is studying one of the most abstract questions of the theory of functions to determine whether on a given Riemann surface there always exists a function admitting of given singularities. What does the celebrated German geometer do? He replaces his Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribution of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation.

    So in a sense the intuitive leap is a necessary feature of how something written mathematically can be part of the analogy written as a visual attribute, from such modelling?


  30. What a wonderful coincidence! I recently had a discussion in a course on the history of natural sciences on exactly this topic. I defended the opinion that metaphors are all bad and should be shunned at all times. Unfortunately, the social scientists in the room did not agree :)

  31. I generally seem to be missing the mark here. Nevertheless, it occurs to me that metaphor itself may be a metaphor for a deeply rooted, fundamental characteristic of nature. One thing being like another need not necessarily be a given property, yet it is. Can we even entertain the question of what makes it so?

  32. Unfortunately many of today scientists seem to be ignoring the lessons taught to us by Einstein.  They chose to look for reality only in terms of abstract mathematics instead of
    the physical imagery given to us by the reality of what we can see and touch.

  33. Regarding my recent blurbs about emergence of scientific theories you may find relevant this: “In physics, the complications all condense into an emergent, simpler description,” Sethna said. “In many other fields, this condensation is hidden – but it’s still true that many details don’t matter.”

  34. “A good sentence can tether a man for a thousand years” – spoken by the sympathetic Bedouin character in a Hollywood film.

    Perhaps it is only other people’s metaphor that theoretical physicists needs worry about. The moment of first apprehension is likely kindred to metaphoric; only freshly made.

    Perhaps physical mathematics and metaphor will forever be the “Odd Couple,” the one rigorous and precise -- the other more flamboyant and freewheeling. Yet, they are forced to cohabitate the same mind and drink at the same well.

    I have wondered… if you took someone with an Einstein intellect, taught them all of modern physics with mathematical rigor, but raised him or her in a laboratory without any contact with the natural world --- would they eventually “discover” the possibility of rainbows? I guess it is possible that this question has been formally dealt with.

    “The sea knows only the bottom of the boat” – Carl Sandburg

    So I wonder if there is some larger organizing principal that is implicit in our physics, but not clearly drawn. (Or yes, perhaps I prefer a good story to actual science.)

    Perhaps metaphor can perform a simple sum where mathematics would struggle. What do you get when you add change to constraint?

    What if the quantum realm is not foundational or determinant, what if it’s simply a medium of possibility in which the crystal of reality takes form?

    Words are tricky, clarity is not a given.


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