Beauty ≠ truth

Scientists prize elegant theories, but a taste for simplicity is a treacherous guide. And it doesn’t even look good

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Theoretically beautiful; geometrically pruned trees, Leer, Germany. Photo by Karl Johaentges/Getty

Theoretically beautiful; geometrically pruned trees, Leer, Germany. Photo by Karl Johaentges/Getty

Philip Ball is a British science writer, whose work appears in Nature, New Scientist and Prospect, among others. His latest book is Serving the Reich: The Struggle for the Soul of Physics Under Hitler (2013).

Albert Einstein’s theory of general relativity is a century old next year and, as far as the test of time is concerned, it seems to have done rather well. For many, indeed, it doesn’t merely hold up: it is the archetype for what a scientific theory should look like. Einstein’s achievement was to explain gravity as a geometric phenomenon: a force that results from the distortion of space-time by matter and energy, compelling objects – and light itself – to move along particular paths, very much as rivers are constrained by the topography of their landscape. General relativity departs from classical Newtonian mechanics and from ordinary intuition alike, but its predictions have been verified countless times. In short, it is the business.

Einstein himself seemed rather indifferent to the experimental tests, however. The first came in 1919, when the British physicist Arthur Eddington observed the Sun’s gravity bending starlight during a solar eclipse. What if those results hadn’t agreed with the theory? (Some accuse Eddington of cherry-picking the figures anyway, but that’s another story.) ‘Then,’ said Einstein, ‘I would have been sorry for the dear Lord, for the theory is correct.’

That was Einstein all over. As the Danish physicist Niels Bohr commented at the time, he was a little too fond of telling God what to do. But this wasn’t sheer arrogance, nor parental pride in his theory. The reason Einstein felt general relativity must be right is that it was too beautiful a theory to be wrong.

This sort of talk both delights today’s physicists and makes them a little nervous. After all, isn’t experiment – nature itself – supposed to determine truth in science? What does beauty have to do with it? ‘Aesthetic judgments do not arbitrate scientific discourse,’ the string theorist Brian Greene reassures his readers in The Elegant Universe (1999), the most prominent work of physics exposition in recent years. ‘Ultimately, theories are judged by how they fare when faced with cold, hard, experimental facts.’ Einstein, Greene insists, didn’t mean to imply otherwise – he was just saying that beauty in a theory is a good guide, an indication that you are on the right track.

Einstein isn’t around to argue, of course, but I think he would have done. It was Einstein, after all, who said that ‘the only physical theories that we are willing to accept are the beautiful ones’. And if he was simply defending theory against too hasty a deference to experiment, there would be plenty of reason to side with him – for who is to say that, in case of a discrepancy, it must be the theory and not the measurement that is in error? But that’s not really his point. Einstein seems to be asserting that beauty trumps experience come what may.

He wasn’t alone. Here’s the great German mathematician Hermann Weyl, who fled Nazi Germany to become a colleague of Einstein’s at the Institute of Advanced Studies in Princeton: ‘My work always tries to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.’ So much for John Keats’s ‘Beauty is truth, truth beauty.’ And so much, you might be tempted to conclude, for scientists’ devotion to truth: here were some of its greatest luminaries, pledging obedience to a different calling altogether.

Was this kind of talk perhaps just the spirit of the age, a product of fin de siècle romanticism? It would be nice to think so. In fact, the discourse about aesthetics in scientific ideas has never gone away. Even Lev Landau and Evgeny Lifshitz, in their seminal but pitilessly austere midcentury Course of Theoretical Physics, were prepared to call general relativity ‘probably the most beautiful of all existing theories’. Today, popularisers such as Greene are keen to make beauty a selling point of physics. Writing in this magazine last year, the quantum theorist Adrian Kent speculated that the very ugliness of certain modifications of quantum mechanics might count against their credibility. After all, he wrote, here was a field in which ‘elegance seems to be a surprisingly strong indicator of physical relevance’.

We have to ask: what is this beauty they keep talking about?

Some scientists are a little coy about that. The Nobel Prize-winning physicist Paul Dirac agreed with Einstein, saying in 1963 that ‘it is more important to have beauty in one’s equations than to have them fit experiment’ (how might Greene explain that away?). Yet faced with the question of what this all-important beauty is, Dirac threw up his hands. Mathematical beauty, he said, ‘cannot be defined any more than beauty in art can be defined’ – though he added that it was something ‘people who study mathematics usually have no difficulty in appreciating’. That sounds rather close to the ‘good taste’ of his contemporaneous art critics; we might fear that it amounts to the same mixture of prejudice and paternalism.

Given this history of evasion, it was refreshing last November to hear the theoretical physicist Nima Arkani-Hamed spell out what ‘beauty’ really means for him and his colleagues. He was talking to the novelist Ian McEwan at the Science Museum in London, during the opening of the museum’s exhibition on the Large Hadron Collider. ‘Ideas that we find beautiful,’ Arkani-Hamed explained, ‘are not a capricious aesthetic judgment’:
It’s not fashion, it’s not sociology. It’s not something that you might find beautiful today but won’t find beautiful 10 years from now. The things that we find beautiful today we suspect would be beautiful for all eternity. And the reason is, what we mean by beauty is really a shorthand for something else. The laws that we find describe nature somehow have a sense of inevitability about them. There are very few principles and there’s no possible other way they could work once you understand them deeply enough. So that’s what we mean when we say ideas are beautiful.

Does this bear any relation to what beauty means in the arts? Arkani-Hamed had a shot at that. Take Ludwig van Beethoven, he said, who strove to develop his Fifth Symphony in ‘perfect accordance to its internal logical structure’.

it is precisely this that delights mathematicians in a great proof: not that it is correct but that it shows a tangibly human genius

Beethoven is indeed renowned for the way he tried out endless variations and directions in his music, turning his manuscripts into inky thickets in his search for the ‘right’ path. Novelists and poets, too, can be obsessive in their pursuit of the mot juste. Reading the novels of Patrick White or the late works of Penelope Fitzgerald, you get the same feeling of almost logical necessity, word by perfect word.

But you notice this quality precisely because it is so rare. What generally brings a work of art alive is not its inevitability so much as the decisions that the artist made. We gasp not because the words, the notes, the brushstrokes are ‘right’, but because they are revelatory: they show us not a deterministic process but a sensitive mind making surprising and delightful choices. In fact, pure mathematicians often say that it is precisely this quality that delights them in a great proof: not that it is correct but that it shows a personal, tangibly human genius taking steps in a direction we’d never have guessed.

‘The things that we find beautiful today we suspect would be beautiful for all eternity’: here is where Arkani-Hamed really scuppers the notion that the kind of beauty sought by science has anything to do with the major currents of artistic culture. After all, if there’s one thing you can say about beauty, it is that the beholder has a lot to do with it. We can still find beauty in the Paleolithic paintings at Lascaux and the music of William Byrd, while admitting that a heck of a lot of beauty really is fashion and sociology. Why shouldn’t it be? How couldn’t it be? We still swoon at Jan van Eyck. Would van Eyck’s audience swoon at Mark Rothko?

The gravest offenders in this attempted redefinition of beauty are, of course, the physicists. This is partly because their field has always been heir to Platonism – the mystical conviction of an orderly cosmos. Such a belief is almost a precondition for doing physics in the first place: what’s the point in looking for rules unless you believe they exist? The MIT physicist Max Tegmark now goes so far as to say that mathematics constitutes the basic fabric of reality, a claim redolent of Plato’s most extreme assertions in Timaeus.

But Platonism will not connect you with the mainstream of aesthetic thought – not least because Plato himself was so distrustful of art (he banned the lying poets from his Republic, after all). Better that we turn to Immanuel Kant. Kant expended considerable energies in his Critique of Judgment (1790) trying to disentangle the aesthetic aspects of beauty from the satisfaction one feels in grasping an idea or recognising a form, and it does us little good to jumble them up again. All that conceptual understanding gives us, he concluded, is ‘the solution that satisfies the problem… not a free and indeterminately final entertainment of the mental powers with what is called beautiful’. Beauty, in other words, is not a resolution: it opens the imagination.

Physicists might be the furthest gone along Plato’s trail, but they are not alone. Consider the many chemists whose idea of beauty seems to be dictated primarily by the molecules they find pleasing – usually because of some inherent mathematical symmetry, such as in the football-shaped carbon molecule buckminsterfullerene (strictly speaking, a truncated icosahedron). Of course, this is just another instance of mathematics-worship, yoking beauty to qualities of regularity that were not deemed artistically beautiful even in antiquity. Brian Greene claims: ‘In physics, as in art, symmetry is a key part of aesthetics.’ Yet for Plato it was precisely art’s lack of symmetry (and thus intelligibility) that denied it access to real beauty. Art was just too messy to be beautiful.

In seeing matters the other way around, Kant speaks for the mainstream of artistic aesthetics: ‘All stiff regularity (such as approximates to mathematical regularity) has something in it repugnant to taste.’ We weary of it, as we do a nursery rhyme. Or as the art historian Ernst Gombrich put it in 1988, too much symmetry ensures that ‘once we have grasped the principle of order… it holds no more surprise’. Artistic beauty, Gombrich believed, relies on a tension between symmetry and asymmetry: ‘a struggle between two opponents of equal power, the formless chaos, on which we impose our ideas, and the all-too-formed monotony, which we brighten up by new accents’. Even Francis Bacon (the 17th-century proto-scientist, not the 20th-century artist) understood this much: ‘There is no excellent beauty that hath not some strangeness in the proportion.’

Perhaps I have been a little harsh on the chemists – those cube- and prism-shaped molecules are fun in their own way. But Bacon, Kant and Gombrich are surely right to question their aesthetic merit. As the philosopher of chemistry Joachim Schummer pointed out in 2003, it is simply parochial to redefine beauty as symmetry: doing so cuts one off from the dominant tradition in artistic theory. There’s a reason why our galleries are not, on the whole, filled with paintings of perfect spheres.

Why shouldn’t scientists be allowed their own definition of beauty? Perhaps they should. Yet isn’t there a narrowness to the standard that they have chosen? Even that might not be so bad, if their cult of ‘beauty’ didn’t seem to undermine the credibility of what they otherwise so strenuously assert: the sanctity of evidence. It doesn’t matter who you are, they say, how famous or erudite or well-published: if your theory doesn’t match up to nature, it’s history. But if that’s the name of the game, why on earth should some vague notion of beauty be brought into play as an additional arbiter?

Because of experience, they might reply: true theories are beautiful. Well, general relativity might have turned out OK, but plenty of others have not. Take the four-colour theorem: the proposal that it is possible to colour any arbitrary patchwork in just four colours without any patches of the same colour touching one another. In 1879 it seemed as though the British mathematician Alfred Kempe had found a proof – and it was widely accepted for a decade, because it was thought beautiful. It was wrong. The current proof is ugly as heck – it relies on a brute-force exhaustive computer search, which some mathematicians refuse to accept as a valid form of demonstration – but it might turn out to be all there is. The same goes for Andrew Wiles’s proof of Fermat’s Last Theorem, first announced in 1993. The basic theorem is wonderfully simple and elegant, the proof anything but: 100 pages long and more complex than the Pompidou Centre. There’s no sign of anything simpler.

It’s not hard to mine science history for theories and proofs that were beautiful and wrong, or complicated and right. No one has ever shown a correlation between beauty and ‘truth’. But it is worse than that, for sometimes ‘beauty’ in the sense that many scientists prefer – an elegant simplicity, to put it in crude terms – can act as a fake trump card that deflects inquiry. In one little corner of science that I can claim to know reasonably well, an explanation from 1959 for why water-repelling particles attract when immersed in water (that it’s an effect of entropy, there being more disordered water molecules when the particles stick together) was so neat and satisfying that it continues to be peddled today, even though the experimental data show that it is untenable and that the real explanation probably lies in a lot of devilish detail.

I would be thrilled if the artist were to say to the scientist: ‘No, we’re not even on the same page’

Might it even be that the marvellous simplicity and power of natural selection strikes some biologists as so beautiful an idea – an island of order in a field otherwise beset with caveats and contradictions – that it must be defended at any cost? Why else would attempts to expose its limitations, exceptions and compromises still ignite disputes pursued with near-religious fervour?

The idea that simplicity, as distinct from beauty, is a guide to truth – the idea, in other words, that Occam’s Razor is a useful tool – seems like something of a shibboleth in itself. As these examples show, it is not reliably correct. Perhaps it is a logical assumption, all else being equal. But it is rare in science that all else is equal. More often, some experiments support one theory and others another, with no yardstick of parsimony to act as referee.

We can be sure, however, that simplicity is not the ultimate desideratum of aesthetic merit. Indeed, in music and visual art, there appears to be an optimal level of complexity below which preference declines. A graph of enjoyment versus complexity has the shape of an inverted U: there is a general preference for, say, ‘Eleanor Rigby’ over both ‘Baa Baa Black Sheep’ and Pierre Boulez’s Structures Ia, just as there is for lush landscapes over monochromes. For most of us, our tastes eschew the extremes.

Ironically, the quest for a ‘final theory’ of nature’s deepest physical laws has meant that the inevitability and simplicity that Arkani-Hamed prizes so highly now look more remote than ever. For we are now forced to contemplate no fewer than 10500 permissible variants of string theory. It’s always possible that 10500 minus one of them might vanish at a stroke, thanks to the insight of some future genius. Right now, though, the dream of elegant fundamental laws lies in bewildering disarray.

An insistence that the ‘beautiful’ must be true all too easily elides into an empty circularity: what is true must therefore be beautiful. I see this in the conviction of many chemists that the periodic table, with all its backtracking sequences of electron shells, its positional ambiguities for elements such as hydrogen and unsightly bulges that the flat page can’t constrain, is a thing of loveliness. There, surely, speaks the voice of duty, not genuine feeling. The search for an ideal, perfect Platonic form of the table amid spirals, hypercubes and pyramids has an air of desperation.

Despite all this, I don’t want scientists to abandon their talk of beauty. Anything that inspires scientific thinking is valuable, and if a quest for beauty – a notion of beauty peculiar to science, removed from art – does that, then bring it on. And if it gives them a language in which to converse with artists, rather than standing on soapboxes and trading magisterial insults like C P Snow and F R Leavis, all the better. I just wish they could be a bit more upfront about the fact that they are (as is their wont) torturing a poor, fuzzy, everyday word to make it fit their own requirements. I would be rather thrilled if the artist, rather than accepting this unified pursuit of beauty (as Ian McEwan did), were to say instead: ‘No, we’re not even on the same page. This beauty of yours means nothing to me.’

If, on the other hand, we want beauty in science to make contact with aesthetics in art, I believe we should seek it precisely in the human aspect: in ingenious experimental design, elegance of theoretical logic, gentle clarity of exposition, imaginative leaps of reasoning. These things are not vital for a theory that works, an experiment that succeeds, an explanation that enchants and enlightens. But they are rather lovely. Beauty, unlike truth or nature, is something we make ourselves.

Philip Ball will be appearing in London on July 7 to talk about this article. Discounted tickets are available for Aeon readers. To be notified when tickets go on sale click here. This event is organised by The Browser in association with Aeon and Prospect Magazine.

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  • natcase

    And this is a beautiful article. That last sentence really sums up a great deal. The same could be said for theories: however elegant or beautiful (or maddeningly complex and full of caveats) they are, they are not the things they describe, but a human explanation of them.

    One of the ways scientific philosophies consistently overreach is by assuming that an explanation of a general behavior must therefore explain all specific instances: the fallacy that the universe is governed by laws that we must discover. As humans whose societies at all scales are governed by laws, this is a natural (even beautiful) projection. It is very hard to wrap a human mind around the idea that specific instances seemingly always conform to our explanations not because of law but because the statistical likelihood of that behavior occurring is very very high. In other words, exceptions are possible but exceedingly rare, and in most case are cancelled out by the overwhelming tide of conforming behaviors. This somehow feels un-beautiful, partly because it seemingly legitimizes monsters and miracles and other holes in the orderly fabric of space-time.

    Beauty is imperfect because we are imperfect, and we are imperfect because our model of perfection is too simplistic.

    • john8

      So you believe in miracles?

      • natcase

        At the scale of human perception and interaction, the kinds of exceptions I'm talking about — one particle in a very very large number does not behave in the usual manner—are so small as to me effectively cancelled out by all the other particles acting the usual way. So, no, I don't believe in the kind of magics and miracles that make our eyes widen when they happen in stories. But I do think there is an important distinction between describing a general behavior and describing a specific instance or event. And I do like magical stories, a lot. And I'm very interested in why they seem to be so important for so many people.

  • MaceKelly

    I think of the old saw, "Beauty is in the eye of the beholder", for what ever, if any, that adds. BUT, I think that quip is the essense of what I would say, which is, these theories, and beautiful things, trigger chemical reactions in the brain, the pleasure drugs, on a deep scale within the brain, creating neuron paths that triggers great drugs. I have often sat and looked at a 'beautiful' sun set or forest valley setting and wondered why it 'seemed' so beautiful. It was abvious that the scene triggered or was in haromny with some mental functions, or brain structures. My dog might look at that scene and have no brain reward chemical floods, but something else might. It is all a closed loop system. All the beauty is in the eye (brain chemistry) of the beholder.

  • joe average

    For a scientist, it is truth that is beautiful.

    • susan

      as a mathematician, beauty is a great indicator of deep mathematics.
      we have a good sense of it.

      as a mathematical physicist, i agree with dick feynman " if your theory disagrees with experiment, no matter how beautiful, it is wrong."

    • ApathyNihilism

      Not necessarily. The truth can be very ugly.

  • annie

    As an artist the photo on this article really bothered me. How exactly does the meticulous pruning of trees translate to simplicity? Superficially I see how it's supposed to work, but conceptually it's horribly messy, and actually works against the author's argument, as the inelegant approach results in a damn ugly garden.

  • Doc c

    Beauty is an emotional response. It is real because our emotions are real, and grounded in our natural environment. It serves an evolutionary purpose. That we have such a response to our attempts to master nature by deciphering its underlying operational mechanisms demonstrates just how transcendent science really is. Nature, as science would have it, is meaningless. purposeless, random and deterministic. Beauty has no place in such a Universe. In that universe we have no more meaning or value than rocks. We are just rocks that can talk. But evolution has given us an urge to transcend our context as evolving organisms in that universe. It is time for scientists to accept their fate as such transcendent creatures.

  • Ken Weiss

    This is a very appealing article and well-written. It's compelling if one takes it in the modern context, that is, the context of contemporary thinking. But it is, I think, anthropologically quite naive and hence loses some of its implied cosmic relevance. Tastes for beauty vary around the world and, perhaps more to the point, with in a given culture over very short time periods. What one generation thinks is beautiful another thinks as ugly or garish.

    From this point of view, ideas such as symmetry, mathematical analytic power (as opposed, say, to approximations that look more neat than they are, such as by partial power series or statistical probability statements), are perhaps very relevant to the pursuit of science in our own particular time. They may be less so in a future time, even in science. You deal with the issue of symmetry vs asymmetry, which touches on the issues, if they are matters of esthetic preference.

    From Aristotle for 2000 or so years, the perfect crystal spheres on which the stars were embedded, or the Pythagoreans' 'music of the spheres', were beauty personified if anything was. But when Galileo disabused them of this sort of perfection, some other sort of beauty had to take its place. Following on Galileo's notions, Newton's god-given laws of Nature gave the cosmos a beautiful tight, mathematical nature. Einstein may have had the same sort of implicit objective in, say, making sure that relativity applied to every frame of reference. And then there's the beauty of the Higgs confirmation.

    And here we are only discussing western science and culture; did not the
    other world's populations' cosmologies have their own type of 'beauty'?

    To me this is about our current concept of elegance rather than beauty. 'Elegance' can perhaps be defined in less culture-context dependent terms, though even it may not have universal aesthetic appeal, which most people would think 'beauty' refers to. It is an interesting property of Nature, whether or not we delve into questions such as whether there even is one Truth, or whether metaphoric truths have beauty even if they're not empirically accurate (many would argue the Bible or Koran give cosmic explanations of untold beauty).

  • Paul Davis

    I'd have thought, to be beautiful, each element in the gravitational equations would have been raised to the fourth power. Then it would be perfectly symmetric in every way. It isn't.

  • Derek Roche

    I think you get closest to the beautiful truth with the phrase internal logical necessity, with the emphasis on 'internal'. Internal logical necessity.

    This is what the great literary and musical compositions achieve and they do so without necessarily referring to anything external. Music especially is 'art for art's sake' and all art, so they say, aspires to the condition of music.

    The same effect can be achieved in mathematics with a simple, recursive iteration such as the Mandelbrot set which is contained within, i.e. internal to, the unit circle in the complex plane. Most people will agree it's a beautiful thing yet it's also said to be the most complex object in mathematics.

    If the world is all that is the case, uncaused by external agency, surely it follows that what is both true and beautiful about it is a function of internal logical necessity.

    • Guest

      The Mandelbrot set is actually contained within the circle of radius 2.

      • Derek Roche

        Hmm, yeah, you're right. I stand corrected. Then again, here's one that is contained within the unit circle:

  • SmoMo

    I feel that this article is confused.
    The beauty of a scientific theory is relative to other theories that describe the same phenomena.

    Just like, if we both created a painting to describe the same historic event.
    The beauty is never between our painting in relation to the actual event, but only in relation to each other paintings of the same theme.

    Who would argue that, for example, Picasso's Guernica is more beatiful than the actual bombing of Guernica was?
    What would that even mean?

    So the Truth that the article talks about, is really just the totality of the evidence collected, and there is no difficulty in describing this Truth 100% accurately, because you need only repeat the evidence a 2nd time.

    But just as the Bombing of Guernica could be 100% described by a list of all the atoms ( and other little particles ) and the trajectories that they took, it would not be elegant, it would not be very simple, and it wouldn't be considered a beautiful description.

    A better description is one in which the details are replaced with vague smooth forms, not because the details are considered unreal, but because the details are considered less relevant than the larger, general forms.

  • Bill

    This is an interesting article, and an enjoyable read. However, the author failed to mention the other Einstein quote: "Everything should be made as simple as possible, but not simpler." I think this quote explains much more about the process of developing scientific theories than anything else in the article.

    • ApathyNihilism

      That's the Ockham's Razor approach, which the author does include.

      • Craig Axford

        The author included Occam's Razor, but he appears to think it states the simplest explanation tends to be the right one. What it states is what Einstein is quoted as saying above: the simplest explanation that also accurately describes the phenomenon in question tends to be the right one. That might still be an extremely complex explanation compared to other possible explanations. Simplicity is only part of the equation, but it is being presented here as being the only thing that matters.

    • dbjm

      Agreed. But even though Einstein seemed to emphasize simplicity above all else, simplicity by itself is not the only valued characteristic of scientific theory: Generality and especially falsifiability are also considered highly desirable. The culture of science largely revolves around the effort to develop theories having these three characteristics. In other words, scientists seek to explain as much as possible (generality) with as little as possible (simplicity), and in a manner that is capable of being disconfirmed (falsifiability).

    • Jimmy

      And there's the other-other Einstein quote -

      "The most beautiful and profound emotion we can experience is the sensation of the mystical. He to whom this emotion is a stranger, who can no longer wonder and stand rapt in awe, is as good as dead. To know that what is impenetrable to us really exists, manifesting itself as the highest wisdom and the most radiant beauty, which our dull faculties can comprehend only in their primitive forms - this knowledge, this feeling, is at the center of true religion."

      Mystical experience seems to be the ultimate in experience of beauty, attested to by everyone undergoing one - what Einstein's experience of this was, and how he related it to science - I'd like to know.

      • The Day Breaks for Friedrich N

        To Einstein's quote: It's interesting that he used the word 'mystical', because I think one can feel that way just looking at the world, knowing that our senses can only perceive a small portion of it, and thinking about what it would be like to directly see in the UV spectrum or sense electric charge, as some creatures can. I would call that 'A heightened awareness of our perceptual limits/distortions, and all that might be unpercievable by us, but it seems to be the same feeling.

  • John R. Moran

    I think a perfect illustration of this is the reaction to Thomas Piketty's "Capital in the Twenty-First Century."

    Piketty has done us a great service in gathering painstakingly detailed and novel data to show an incredibly historical relationship. He has extrapolated from this to a very elegant predictive theory.

    But I worry that the very simplicity, clarity, and "newness" of the theory (trumping so much messy history of economic thought) has got a lot of pundits head over heels, and possibly overlooking some weaknesses and contrary explanations.

  • prior probability

    Is bayes' theorem beautiful?

    • Guest

      Not really, it's a rather dull theorem in itself. An obvious corollary of conditional probability.

      • prior probability

        ... so a theory doesn't have to be beautiful to be helpful!

        • The Day Breaks for Friedrich N

          People do get into trouble using it where it doesn't apply, and assuming all kinds of things are 'independent of' each other when it's not the case, however. Maybe carried away by the simplicity :)

  • Marcos

    There is nothing like a casual reading of any anatomy, physiology, or biochemistry book to shatter down to pieces the illusion of beauty in natural processes. And even in those disciplines authors go great lengths to make it look as a beautiful clockwork piece of art when is really what I like to call "a mess that works well enough to perpetuate the structure in the fourth dimension". Great article. The last paragraph toped it off beautifully (pun very much intended).

  • Kaushik Kalyanaraman

    It appears to me that the main premise of the article — Scientists essentially seek beauty in constructing theories for how nature works — is a gross simplification.

    At some level, and to borrow from the words of Neil Turok, the Physics of the very large, relativity, and the Physics of the very small, quantum mechanics, can be thought of as being "simple" theories. Yet both these elegant — "beautiful" — mathematical theories are incomplete for not being able to provide explanations for a variety of problems within their own regimes. In my gently-informed-yet-non-physicist’s understanding, the apparent "fine tuning" problems that come in many flavors leading to questioning why the large-scale universe perhaps even exists, and the inability of the Standard Model to provide any reasonable methodology for explaining dark matter are two such examples. And, the middle regime that either arises up, or reduces down to, from the ends is in fact so utterly complicated that to characterize some of its structures — as the author alludes to in the classification of chemical elements or a theory for biological evolution, for example — as being beautiful in some universal sense is perhaps tenuous.

    However, this does not preclude scientists, who devote their lives to pursuits within a narrow sliver in a tiny fraction of the enormous behemoth that is Science, from finding beauty and perhaps even inner meaning in their own field or work. In the same style of the author's rather unfortunate nitpickings of Einstein and others' views on beauty in Science, it can be said that in human society, every mother finds her own child to be beautiful, and to hold such a position up to scrutiny would be unwise.

    Mathematics, on the other hand, is open to interpretations of subjective beauty simply because it is not bound, nor does it particularly claim to as far as I can tell, to provide any explanations for the objective world. The beauty in math may find its way into theories for reality but the Platonic world is a place that is anything unlike its physical counterpart. In my opinion, this is the aspect that seems to have been lost in the author's attempt to break up the supposed equivalence between truth and beauty in Science.

    To conclude, it is also somewhat saddening that Philip seems to have missed quoting Richard Feynman's views on both the nature of simplistic theories of nature (circularity intended), and the apparent divide between Art and Science. On the former, Feynman states, in the context of what he is looking for from Physics (or science more broadly):

    "I'm just looking to find out more about the world and if it turns out there is a simple ultimate law which explains everything, so be it, that would be very nice to discover. If it turns out it's like an onion with millions of layers and we're just sick and tired of looking at the layers, then that's the way it is, but whatever way it comes out its nature is there and she's going to come out the way she is, and therefore when we go to investigate it we shouldn't predecide what it is we're trying to do except to find out more about it." [The Pleasure of Finding Things Out]

    I suspect that it won't be a stretch to assume that Feynman's take can also be extended to the question of whether beauty is a prerequisite or not. Also, in that very same interview, Feynman suggests that Science and scientific theories by themselves only add to the perhaps human constructed sense of aesthete, and not subtract from it. Thus, in single discussion, Feynman provides the counterpoint to Philip Bell that may invalidate his premises.

  • Vara Sue Tamminga

    I have always wondered about "moral truth" in relation to physical beauty. It is very curious to me that someone like Helen Keller or Eleanor Roosevelt or Gandhi can seem very beautiful whereas an extremely attractive, physically symmetrical model or movie star with stunning good looks who is known to be mean or petty can seem ugly. If beauty or truth were only a matter of symmetry or harmony, then Elizabeth Taylor or Marylyn Monroe might win the beauty pageant prize. But actually, the self centeredness of movie stars or the roles they play often seems to interfere with our perception of beauty so that someone who is rather plain physically, like Gandhi, but who possesses remarkable inner worth, can seem actually more beautiful than a movie star. When we discuss Physics as a realm of truth or beauty, perhaps we are staring into a cosmic value of goodness as well or into Plato's ideal forms. My field is poetry. Certainly it is not the symmetry of rhyme or meter that makes a poem beautiful otherwise nursery rhymes would be more beautiful than Shakespeare who always keeps us guessing, keeps the meaning of his poem or play wedded to the mystery of human experience. We call such writers profound; the meaning of poetry is what makes its physical form beautiful.
    If for instance, we discovered that Einstein were a mass murderer, the beauty of his idea would in some strange way change. So I think that there is a moral dimension to beauty which is not as simple as symmetry, but it does approach truth which we emotionally feel has something to do with goodness, with love perhaps, with our desire that the universe have a loving, goodwill that we might call God. Otherwise, I don't understand why I feel that Eleanor Roosevelt is more beautiful than Elizabeth Taylor. Of course, when inner beauty is also combined with physical beauty, as in the paintings and sculpture of great masterpieces, then we are at the gate of some New Jerusalem where goodness, truth, and beauty together form heaven. I wonder if the Mona Lisa, for instance, is so beautiful precisely because we cannot decipher her inner state of mind. We stand continually unsure if she is a beautiful woman like DaVinci's Madonnas or angels or if she is more unfair or shallow than a saint. She lacks the haunting perfection of DaVinci's Virgin Mary or the darker more mysterious face of his St. Anne, but his Mona Lisa is strikingly complex, unsettling, clearly not superficial but also not merely perfect. Wonderful questions. I have known some very beautiful people in my life, but I find it so strange how the cruelty or indifference of so called pretty people makes them so unattractive. They lack truth. And I sense that Keats was right when he married truth and beauty.

  • Mayank Mandava

    I feel this article is way too unfair to the scientists. I can't help but get the impression that the author thinks of them as doddering little Hercules Poirots who have no interest in *real* beauty but only the cold symmetry of logic. This is not what the scientist means by beauty. There is no "narrow definition" of it they are all clamoring towards, but rather an admission of the same subjective experience of awe and transcendence one feels when looking at anything that is described as beautiful by the observer. "Beauty lies in the eyes of the beholder" holds just as true for scientists as it does for artists. And just like Beauty, and unlike Truth or Nature, Mathematics is another thing we make ourselves.

    What is Beauty? It's a quality possessed by some object (weather tangible or not) that evokes from it's beholder *something*. What that thing is will lead to a circular definition. Just like Love, or Happiness. It's a personal relationship between the object and the observer and we have to believe them when they say they find something beautiful, because what else can we do? But one suspects that both artistic and scientific beauty have a relationship with Truth. They both speak a Truth that we have been grasping at but have so far failed to see. "Everything should be made as simple as possible, but not simpler," said Einstein, which agrees rightly with your asserting that beauty vs complexity is U shaped. It's the very co-incidence of Truth in the abstract (in the theory or the painting) with the Truth in the concrete (the experiment or the human condition) that makes an object Beautiful. There is no conflict here. Your premise, that scientific beauty is always an assertion of simplicity, is simply flawed.

  • Roger

    I can't comment on the science part, but the art part is very very good. As is the brief comment about the current Darwinist fashion. Art seeks to satisfy the imagination, which generally prefers something between chaos and perfect order, which "something" gives it something to do.

  • jansand

    It seems to me beauty is what you find delightful. It can be either simple or complex depending on how it affects you. A perfect mathematical form delights me but that delight is not permanent and I could find it uninteresting after a while. A piece of mathematics that carries implications through a wide area of mathematics and simplifies understanding f how they all are related probably could be considered beautiful. As in any scientific discipline. The periodic table in chemistry may have ts complexities but it does indicate a simple relationship between all chemical elements and points to how they differ and why and where they originated and discoveries about their relationships are made clear through the table and point the way to new discoveries. Time and space are likewise clarified through Einstein's theories and perform the same function in revealing many avenues in science to explore.

    Likewise, in the other arts (and I consider science and mathematics as branches of the arts) revelations in insights as to how older rules of observation and skill in manipulation of the elements of the discipline may be extended or changed in a useful way or combined with surprising elements can evoke delight and a sense of beauty. This evocation is an event in itself and whether it is made by a saint or a scoundrel does not make it repellent or more beautiful.

    One must be careful about the understanding of simplicity. What is simple for one mind may well be totally mysterious or ugly to another. Minds differ a great deal.

  • john8

    Ah but there are a few other trees in this grove you should be barking up.

    I seem to be asserting that beauty, as well as consensus, agenda, and nonconformity trump experience come what may.

  • ApathyNihilism

    I find Einstein's General Relativity ugly and silly, at least as it is currently described. Why should we claim that gravity bends space-time, rather than simply claim that gravity affects processes in space and time? To bend space, it would have to be bent relative to some other (more absolute space); to bend time, it would have to be bent relative to some other time. It makes more sense to describe processes, or better yet, their perception, as being relative to the observer.

    • Philip C . Lehar

      It's magic that it should follow the shortest path. But it's intuitive.

    • Guest

      Because it is more ugly and silly to claim that gravity affects processes in space and time than it is to say that gravity bends space-time. Claiming that space-time itself is not flat and that this curvature is caused by mass/energy is actually a neater way of describing the reality. Relativity does give a framework to describe processes relative to a particular observer but it also tells us what aspects of those processes remain invariant when viewed by another observer.

  • ApathyNihilism

    I also prefer monochromes to landscapes, and minimalist music to "Eleanor Rigby".

  • Fredrik Vegstein

    This has little to do with the overall point of the artical, but I would like to add to the part about Einstein and the experiments proving his theory. He may very well have cared little for the expiriments. I don't now anything about his feelings towards them. However it was Einstein himself who first pointed out that the bending of light around the sun could be observed, if the theory was correct.

    The first attempt at this was made by Erwin Finlay-Fruendlich and William Wallace Campbell in 1914 in Crimea, though since they Fruendlich was arrested as a spy due to being german when the first world war broke out. This was the result of Einstein sending letters to leading astronomers around the world for close to 3 years, urging them to make this measurement. They failed and Einstein was reportedly quite devestated, but there is no way for me to confirm that. Fun fact: had they succeeded he would have been proven wrong becouse of mathematical errors in his theory.
    The story of how the theory of special relativity is an intereresting look at science, and goes on for quite a bit longer then what I wrote above.

    Einstein is said to have had a rather large ego, and that he never even considered the possiblilty he was wrong. Never the less, he knew it had to be proven to get accepted. Like you say he is not here to argue, and perhaps it would have been best to stop there. To say that the man who first asked for the test to prove his theory, was indifferent to said tests, seems to me a rather odd thing.
    However saying so makes your "theory" look "cleaner". Some might even say "beautiful". Makes me think of another beutiful thing: irony.

    But perhaps I misunderstood your meaning there. All and all a good read. I enjoyed it.

  • Guest

    Einstein didn't believe the theory was too beautiful to be wrong, he thought it was too beautiful to ignore. Beauty isn't truth, it's beauty and that means something. It isn't a false signpost, it's a benchmark on the path to truth. Otherwise we wouldn't see it as beautiful.

    • lennyharris

      I like what you said.

  • arendt

    Galileo rejected Kepler's planetary ellipsis in favor of the circularity of planetary motion because he found the circle more beautiful. Notions of beauty are, in part, culturally determined. Scientific truths (with a small "t") are not culturally determined, at least in the natural sciences.

  • jansand

    It is important to understand chaos. Like beauty, it is not a state of the universe it is a state of mind. It indicates confusion as to what constitutes observed order. The universe is extremely finely ordered and science, through speculation and observation and confirmation is the process of determining the nature of the orders in operation in an observation. Speculation alone is insufficient since speculation can only try to match a presumed order to observation and that frequently fails. Religion frequently presumes orders that are not confirmed or even are not possible to confirm and demands belief in these speculations nevertheless which frequently causes huge problems. Science has small difficulty in discarding presumed orders when observation indicates a mismatch. Mathematics is not science, it is a discipline of examining precisely different speculative order systems and when a mathematical system is discovered by science to match observation it is very useful for examining further observations and integrating them into a useful theory.

  • Vasco Gama

    Philip Ball,

    It seems to me that something is quite wrong with your conceptions of truth and beauty.

    I would advise you to consider Plato´s "allegory of the cave".

    • The Day Breaks for Friedrich N

      Not for the idea that the things within the world that we call 'beautiful' must all come from an amorphous mass of 'absolute beauty' floating in the aether, I hope! :D

  • Robert Landbeck

    "Beauty, unlike truth or nature, is something we make ourselves." NO, beauty is something we 'imagine' so to maintain a view of ourselves, that like so much theory, may in the end, prove itself not tenable!

  • jansand

    Plato with his confining cave was sure that reality was available through the purity of mere thought alone that performed interplays of acrobatics with consistent abstract symbolisms. That is the fantasy of mathematics which is quite adept at manufacturing all sorts of universes that may play pleasant music on particular themes that have no counterpart in reality at all. For mathematics is not science, it is a rather strict type of science fiction. When delightful parallels in mathematical adventures can be discovered in observing reality, that particular variety may indeed be excellently useful in physics and astronomy and other disciplines tightly bound to observing reality but much of mathematics, like any other artistic occupation, is a delight in self consistent systems that have no counterpart in observations.

    It is important to be aware that we are human animals provided with a set of sense apparatus developed to feed us and keep us aware of saber tooth tigers and to reproduce but not necessarily to analyze the atomic manufacture of elements from basic hydrogen in stars or to comprehend the nature of neurological mental capacities or to speak with bats and dolphins in ultra high frequencies. We have been developing technology to extend our senses but the universe contains far more than we can deal with easily as Earthbound creatures. Such complete basics such as dark matter and energy are fresh novelties and pretty mysterious at the moment and not available by peeks over our shoulders from that mythical cave.

    • Philip C . Lehar

      No idea what you said but it rings true.

  • Peter

    Words are ambiguous. When a chemist reflects on the "beauty" of the periodic table surely they are not referring to the shape of the table and "all its backtracking sequences of electron shells, its positional
    ambiguities for elements such as hydrogen and unsightly bulges that the
    flat page can’t constrain".

    Using the word beauty to describe the periodic table is provocative for this very reason. Such a statement challenges us to look beyond the backtracking sequences of electron shells to see something more profound - something hidden. There are different ways of experiencing beauty, and at least one of them is related to beauty's connection to truth - especially "hidden truths" that are revealed by human genius.

    We experience beauty in multifaceted ways. Any attempt to reduce our experience of beauty to one singular irreducible definition is misguided.

  • Jimmy

    Whenever I hear a scientist telling saying there is beauty in equations, I just get the impression that they really mean is they see a job well done, like a carpenter looking at a well made mortise and tenon joint. More to do with survival pleasure/reward mechanisms.


    I applaud Philip's elegant piece. There is another dimension to physicists obsession with the idea of 'beauty' - the religious dimension of this discourse. Greek philosophers - Plato and especially Pythagoras - associated perfectly regular forms such as circles and spheres with divinity. Platonist cosmology insisted that the structure of the universe must be spherical and that the planets and stars must travel around the earth in circular orbits, because these were the only forms worthy of the divine domain. Platonist/Pythagoreans believed they could describe the structure of the cosmos by a set of idealized ratios which corresponded to the sizes of the various planetary orbits, much as the strings on a lyre can be described by a set of ratios corresponding to its various string lengths. This idea of circles arranged according to idealized ratios is the beginning of a long intellectual tradition that goes by the name 'the harmony of the spheres'. Although we have progressed beyond the spherical-universe model, physicists have never given up the Pythagorean notion of an idealized set of formulae - or perhaps a single perfect formula - that describes everything! Physics talk about 'beauty' remains rooted in archaic ideas about 'divine perfection' and is quintessentially un-modern. Like Newton and Kepler before him, Einstein insisted that 'God' had created a beautiful world - a purely Pythagorean sentiment. But if there was no creator, there is no reason to think any aesthetic precepts pertain. Blaise Pascal offered another view when he wrote: "They say that habit is only second nature, who know but perhaps nature is just the first habit."

  • Abhinav Parihar

    First, idea that science does not favor truth or evidence seems too wrong? When there is evidence, there is no point of subjectivity. Subjectivity arises when there is lack of experimental data or lack of surety in it. And in those cases, why is it wrong to prefer simplicity? In fact, shouldn't that be preferred. If you don't have evidence or confidence in data, then why to describe something you aren't sure of in detail. Why not to prefer simplicity?

    Secondly, you say beauty as used in science is very different from the actual meaning, esp. as used in art. And your main arguments for this:
    1) Beauty in art comes from its randomness, its "messiness", which I think is way off. In many cases, the beauty in art is actually combining the idea with the medium, say painting, in a way to find a match (an order). It is actually the genius of artist that is admired in that beauty.
    2) Simplicity does not guide art, which again I disagree. But even in art, simplicity seems to be a sure guide for beauty. A 100 page description is never more beautiful than a half page poem.

    3) The beauty in science is not truth. But truth sure does not guide beauty in art as well.
    4) Some scientists have been quoted in the article describing beauty, and the author disagrees with their definition. But isn't that so with art. Even the beauty of art is subjective, depends on the person. And different works of art can have different arguments for beauty. And in most likelihood, the definition being quoted might be specific to the person or the work he is referring to.

    The arguments presented might refute some definitions of beauty in science (which is quite true even in case of art), but still fails to give any logical reason for lack of "beauty" in science or its difference from "beauty" in art.

  • Critcop

    We've diverse human endeavors here: science, art, law, etc. I'll admittedly
    oversimplify Kant's point with regard to them all: without
    achieving some cogency in our thoughts, none of these endeavors would
    get us very far, much less pull us together, in coping with otherwise
    chaotic data. Beauty keeps pulling us forward, while (re)focusing our
    attention along the way(s) ...

  • Michael C

    It's interesting to me that the writer contrasts the idea of beauty in science with the idea of beauty in art. Most of the beautiful things in the world have nothing to do with human artifice, and you see them in the garden or at the beach.

  • fynn80

    I believe the author of this article is a bit confused about several things: what is a fundamental physical theory, fundamental v.s. emergent theories, the difference between the definition of beauty in art and in math and so on and so forth.

    A fundamental theory, such as general relativity or the standard model, relies on symmetry principles. I think no one is going to argue that both theories have found extensive experimental confirmations. Their beauty relies in their simplicity, that for a physicist doesn't mean that the theory is mathematically trivial! Everyone who has studied one of the over mentioned theories knows that the math involved is damn difficult. Simplicity means that, starting from a "simple" assumption (the gauge principle) we can obtain many informations out of the theory. It means that, if we take this output and look at some specific regime, we can recover some known theory and/or experimental fact.

    In complex systems, the definition of "fundamental" is more problematic. However, there are regimes where different systems exhibit a universal behaviour, e.g. critical behaviour. The classical theory of phase transitions (Landau's theory) is also based on symmetry principles and has been experimentally verified so many times. It is also simple and beautiful, and in his regime empirically true. Recently, in order to explain more exotic states of matter, a different theory has been proposed, that of Topological order. Also this theory is finding experimental verifications and came out with the idea of "beauty/simplicity".

    If two theories find experimental verification, I will pick up the simpler one, i.e. the one that can explain a phenomena with the smaller number of fitting parameters or assumptions. This is just natural selections, the same that happens for technological products.

    I strongly suggest the author to read "Fearful symmetries" by A. Zee if he is interested in understanding what theorists mean by beautiful, elegant and simple.

    • The Day Breaks for Friedrich N

      'Fearful Symmetries'? I'm just glad he referenced Blake's 'Tyger, Tyger, Burning Bright' poem- and his name is A. Zee, no less! (a little more prosaic than being the Alpha and the Omega) :) Love it...

  • David

    DNA is a beautiful molecule. The way it works is beautiful too.

  • Fred Garvin

    "ON closing this general view of beauty, it naturally occurs, that we should compare it with the sublime; and in this comparison there appears a remarkable contrast. For sublime objects are vast in their dimensions, beautiful ones comparatively small: beauty should be smooth and polished; the great, rugged and negligent; beauty should shun the right line, yet deviate from it insensibly; the great in many cases loves the right line, and when it deviates it often makes a strong deviation: beauty should not be obscure; the great ought to be dark and gloomy: beauty should be light and delicate; the great ought to be solid, and even massive. They are indeed ideas of a very different nature, one being founded on pain, the other on pleasure . . ."

    --Edmund Burke, "A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful" (1757)

    Burke's distinction between the beautiful and the sublime is lost on so many scientists, who after all tend to be neoclasscists. And I understand well the allure of the Platonic solids (I took solid state chemistry from A F Wells many moons ago). But it isn't the whole megillah. One only need look to perceive the asymmetry, chaos, even terror in nature. Taking a different Beethoven example than the Fifth, the musicologist and essayist Richard Tauskin has shown convincingly that it is the aesthetically and morally messy Ninth Symphony that is the quintessentially sublime musical work.

    • Mike

      I agree that the notion of the sublime is good for thinking about this. 'The state of the soul, in which all its motion is suspended' (Burke) is a response to the unbridgeable gulf between the human scale of perception and the ultimate reality of the universe. Beauty speaks to the former; scientific truth is to be found in the latter, with no guarantee that it will accord with human aesthetics.

  • mb2100

    Results showed that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain [...] as the experience of beauty derived from other sources.

    • Jordan Martin

      Thank you.

  • Abhishek Singh Bailoo

    The author obviously didn't cry when he saw e^(i*pi) + 1 = 0 for the first time. That is a pity.

  • Jordan Martin

    The author would be thrilled if artists told scientists that "this beauty of yours means nothing to me". This is a truly peculiar desire, particularly given the fact that many artists employ various terms related to beauty when describing their feelings of awe and wonder toward the scientific understanding of the cosmos, biological life, the brain, etc. They have no problem using the standard concept of beauty for both forms of aesthetic appreciation (nor they do they feel the need to distinguish between distinct forms of beauty), yet the author seems to wish they would. Perhaps the disjunction of scientific and aesthetic beauty intuited by the author is peculiar to his circumstances and is not representative of the concept as it usually employed by both artists and scientists alike.

  • john dagpunar

    If you look at those equations of fundamental physics (for example those propounded by Newton, Einstein, Maxwell, Boltzmann, Dirac, Heisenberg, Schrodinger) that are also borne out by experiment and observation to a high degree, and have stood the test of time, they are mostly beautiful. How do I define beauty in the scientific context? I mean that the equations have an economy of language, have wide generality, and are falsifiable. The historical evidence suggests therefore that beauty may well be a necessary condition for the successful description of fundamental physical phenomena.

    That said, no serious scientist would claim it to be a sufficient condition. When Dirac says "This result is too beautiful to be false; it is more important to have beauty in one's equations than to have them fit experiment", this should not be taken at face value as a belief that a beautiful set of equations is sufficient. Closer examination of the context in which this was said tells us that he was recounting Schrodinger’s experience in holding back publication of a theory of the electron because it was not validated by observation. Dirac was presumably suggesting that a beautiful set of equations can in time be modified (in this case using the realisation that an electron has spin) to achieve the necessary experimental validation. Perhaps Dirac was saying that if a set of (prototype) equations were beautiful it intensifies the resolve to search for the missing pieces of the puzzle that would convert them into a successful theory; and some may empathise with his opinion that a messy theory that matches some data is of inferior value as regards seeking truth (watch out Big Data?); eventually it will be superseded by a an elegant one, revealing much more.

    Dirac was likened to a magician; he would play around with equations and remarkably would come up with mathematical equations that replicated observations and additionally made new predictions. In quantum mechanics, physicists toyed with matrices and created the commutator which predicted the existence of sets of observables that could not be measured simultaneously with certainty. That process of wrestling with equations and succeeding in making a deep seated discovery might be tortuous rather than beautiful (just as in composing a symphony), but to my mind the final equations are beautiful in that the achievement moves the human spirit just as the finished symphony does. However, the two contexts are different in one important respect. The scientific theory was waiting to be revealed whereas the symphony was not. Is this something to do with the infinite number of symphonies (and indeed works of visual art, of literature, mathematical theorems) that can be created, contrasted with a small number of fundamental physical laws waiting to be discovered?

    Another aspect is that we do not for example really know what an electron or proton are, yet the mathematics reveals so much about their behaviour – that is beauty and circumvents the philosophical difficulties associated with the existence and reality of both entities and causality. And then there is the question of explanation and understanding of physical phenomena. In practice, with theories that have stood the test of time, we use familiar terms, intuition, and common sense to do this. Tellingly, Einstein said "Common sense is the collection of prejudices acquired by age eighteen". However, in making ground breaking discoveries, theoretical physicists break away from these prejudices. They have to transcend real world familiarity and are led towards truth through abstraction. That is difficult. In a sense, a mathematical description of a physical phenomenon does not need explanation (notwithstanding that the mathematical derivation does), does not need hand waving arguments, does not rely on familiar terms/ concepts/physical connections and interactions which when we really drill down are frustratingly difficult to explain. If one does follow a reductionist route to its limit one finds that instead of dealing with the supposed reality of entities and processes, that we are left with constructs of them that are mind induced. In essence the derived equations are the definitive extrapolation of that, with a precision that avoids ambiguity and vagueness. It is just mathematics, it is how the world is, and we do the calculation. That is the beauty and elegance of what has now been revealed through an economy of language