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When a theory is beautiful?

  1. Jul 16, 2004 #1
    When a mathematical or a physical theory can be considered as beautiful?
    It is currently admitted that beauty is an important criterion to construct a physical theory, and symmetry is recognised as an element of beauty. How could we measure the aesthetic value of a theory? Or, at experimental level, there are lists of "elegant experiments". When an experiment es elegant?
    I think these questions are related to dilemma "invention vs. discovery" and the Plato's Philosophy.
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  3. Jul 16, 2004 #2


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    Mathematicians have spent (maybe wasted) a lot of time trying to develop a quantitative theory of "beauty" or "elegance" in a theory. It's clear that doing much with little is a big component, the slick trick that solves the big problem. Unifying distant theories is always good, as is breaking new ground.

    For example the Tamiyama-Shimura conjecture is beautiful; is posits a significant and 'nifty' relationship between algebraic curves and number theory. But I don't think the proofs of it are beautiful, because they are nitty-gritty.
  4. Jul 16, 2004 #3
    Beauty as usefulness

    If so, beauty in Science can be reduced to a criterion of efficiency: by minimizing the cost of the experimental method or, alternatively, with an interdisciplinary approach. In the case of an experiment, its simplicity seems a good argument. It would be beauty in the research work. But, how could we measure the intrinsic beauty of a theory with independence of the method involved in its development? Alternatively, Is such question trivial?
  5. Jul 16, 2004 #4


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    I have seen experiments called beautiful on the same basis as theorems. Not only efficiency - getting a big or difficult result from a small input - but above all "cleverness" of the small input. That's what I tried to capture with the word "nifty". Striking, original, ingenious are other words for this quality. Hard to quantify but you know it when you see it.
  6. Jul 16, 2004 #5


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    if it is not in your blood to instinctively and immediately recognize mathematical beauty

    then do not worry about it and do not bother to try to get people to reduce it to false synonyms

    the first person to use a bow and arrow, the first person to sail a boat

    the ability to recognize elegant tools must have been bred into all or most humans

    survival, reproductive success, crafty genes benefit the tribe

    if you can say "ahah!" about a cleverly designed and beautiful physical tool and do not have the same recognition with mathematical ones then it must simply be unfamiliarity with mathematics

    there is no difference (except the context)

    Dante said:

    Considerate le vostre semenza
    Fatti no foste a viver come brutti
    Ma per sequir' virtute e conoscenza

    [consider your breeding
    You were not made to live as animals
    But to understand mathematical beauty]
  7. Jul 17, 2004 #6
    as long as it works...
  8. Jul 17, 2004 #7
    By definition, elegance refers to simplicity and symmetry, especially when discussing scientific theories.

    For example, Einstein's theory of Relativity is widely considered the most beautiful physical theory ever developed. It possess what physicists call supersymmetry (a recursive and global kind of symmetry) and the entire theory boils down to a simple equation: E=MCC.

    Likewise, experiments are considered elegant when they appear to go straight to point and possess a degree of symmetry. For example, Galileo dropping balls off buildings is considered a classic elegant experiment. By simply dropping balls of different weights he created the foundation by which modern physics was eventually created, the so called "weak equivalancy principle."

    Yes, the idea of elegant simplicity is related to Plato's philosophy. If you go back further, it is related to any number of philosophies. Elegant simplicity is simply not a new idea by any stretch of the imagination.
    Last edited: Jul 17, 2004
  9. Jul 17, 2004 #8


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    fascinating discussion

    I dont believe one can define elegant

    It wouldnt be very satisfying to have a special meaning just for "discussing scientific theories" which was different from what the word means in general.
    scientific theories are part of nature like the song of birds
    they come out of people's heads and people are a part of nature

    a gazelle can be elegant
    the way a horse gallops or a fish swims can be elegant
    the way a mathematician describes a proportion in nature can be elegant

    the bones in a woman's face can be elegant

    many words do not have definitions
    and one must just look at how they are used
    that is, they dont have synonyms

    "simple and symmetric" is not a synonym for elegant, at least for me.
    It just does not work, even though it intersects elegant it doesnt capture it.

    sometimes it helps to look at the derivation of a word

    maybe elegant has the same root as elect
    I will look in the dictionary
    sometimes that helps
    Last edited: Jul 17, 2004
  10. Jul 17, 2004 #9


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    elegance comes from the Latin "elegere" to select

    it has the root idea of "correctly chosen"

    and the dictionary gives its first definition as "refined gracefulness"


    that was where my image of the gazelle must have come from

    a gazelle is graceful
    and the body-plan of the gazelle has been refined by evolution
    because she has to outrun predators in open grassland

    so there is this sense of grace and having been correctly selected: a perfect choice
  11. Jul 17, 2004 #10
    Curious, first you say it cannot be defined, which is of course, a definition in itself! Then you proceed to define elegance!

    The definition I gave is a common one among physicists, not the broader definition of the word. The distinction is that physicists are not simply describing common perceptions, but quantifiable physical phenomena. Here is the second half of the definition you so conviently ignored.

    Among other things that scientists have managed to discover, is the impact of symmetry on conventional ideas of beauty and elegance. Of course, a lay person may object to the idea of reducing the concepts to a specific quality that can be quantified, but that is precisely what the sciences endevor to achieve.
  12. Jul 17, 2004 #11


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    but there is no synonym for grace

    none of these words really have synonyms, i suspect

    it does not help to say

    elegant means graceful and graceful, in turn, means fee-fo, and fee-fo means fum, and so on (this doesn't seem to help at all)

    So what I like, wuliheron, is that you are not just trying synonyms but you are showing by example.

    you say Gen Rel is an example of an elegant model

    that is a very well-chosen example IMHO

    I cant recall anything more graceful and elegant that happened in the 20th century (of course my judgement is personal and based on very limited knowledge of course of course).

    But I cant recall anything more graceful and elegant---unless maybe it was the woman I saw dancing at new york civic center ballet under Balanchine around 1960---or the flight of swallows.

    And moreover, Gen Rel was highly original

    (I think you may be wrong about E = MCC. I dont think Gen Rel has anything to do with E = Mc2 which comes in a different
    much less elegant theory that Einstein produced 10 years before Gen Rel)

    General Relativity models gravity (with exquisite accuracy) by means of the geometric shape of the world. It models gravity with shape, instead of force. this is impressively original. I do not think anyone before Einstein thought to do this (but I am not a science history buff and cant speak with certainty)

    Maybe the originality and the remarkable accuracy of its predictions are even more impressive features than the elegance---in the case of General Relativity. Also the fruitfulness: it is telling us now 90 years later about all these things we are finding so exciting: quasars, black holes, gravitational lensing, dark energy, cosmological origins, collapse of stars, pairs of pulsars radiating gravity waves, hypernova gammaray bursts. Nearly everything that seems new and wonderful in astronomy these days seems to be coming out of Gen Rel or have something to do with it.

    So we should not stop with the idea of elegance, we should say
    elegant, graceful, original, and also something like "fertile"----able to go on revealing new wonders even after 90 years, able to keep on suggesting to us things we didnt know and that amazing keep on turning out true---and finally, and perhaps most important, precise.

    Gen rel keeps on being verfied to more and more decimal places of precision. (but who knows, that may stop and some needful correction may be found)

    sorry, these things make my head spin. Anyway it is a very good example of elegance wuliheron. Certain sonnets are also elegant, I forgot about them for a moment.
  13. Jul 17, 2004 #12


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    I humbly beg to disagree with Dictionary.com

    You say that according to Dictionary.com
    elegance has a special meaning of "scientific exactness and precision"

    I dont think that is right.

    We all use words differently I guess so i have a right to disagree and
    Dictionary.com has a right to say that, if that is what he thinks.

    But I have to say that from my point of view Dictionary.com is just plain wrong.

    Elegance and precision are two very different matters.

    Well it looks like we cant discuss anymore because we use words differently and have no means of bridging the difference! thanks for raising a stimulating issue like this!
  14. Jul 17, 2004 #13

    If the aesthetic convictions held by scientists are of value in the development of their theories, such as Steven Weinberg sustains in “Dreams of a final Theory”, the concept of beauty in Science would need be discussed from a philosophical perspective, searching the objectivable aesthetic components of a scientific theory.

    If we can recognize the beauty of a theory (when our genes allow it), but such recognition is subjective and ineffable, as you suggest, the discussion on the beauty of scientific theories lacks of value. We would be ithen in a mystic region. On the contrary, if we can discuss about the aesthetic aspects of a theory, we can made a more rich approach to the historical development of Science.
  15. Jul 17, 2004 #14


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    ryokan I like your style
    although we think differently

    I sense in you a strong optimism
    You take seriously the idea that grace can be objectified
    and since it is part of what has guided the search for mathematical theorems and physical models----since beauty has been a guiding light to some good scientists---this objectified elegance can help in understanding history.

    I dont approve of mysticism---it is a vile ugly business. But I am
    very pessimistic about these things.
    I believe it is of urgent importance to understand history and to
    understand science and to preserve scientific values---objectivity,
    empiricism, openness, transparency, tolerance of difference, discipline
    of rigorous testing etc etc.
    But I am very pessimistic about a lot of things, including the possibility
    of objectifying what i think may be a highly unreliable gut feeling that is maybe half a million years old.

    the sense of beauty is (make no mistake) inconsistent, unreliable, mysterious, entangled with emotion, and all that prickly hairball stuff
    but in a certain department it is all we got
    there is no algoritmic substitute for it
  16. Jul 17, 2004 #15
    Really, we do not think so differently.
  17. Jul 17, 2004 #16
    Last edited: Jul 17, 2004
  18. Jul 18, 2004 #17
    The 'beauty' part might play an important role indeed when 'devising=inventing' (I am not a platonist or a mathematical realist) a scientific hypothesis (I consider only hypotheses which respect the minimum criterions of sciencificity) but what really count is how such hypotheses become part of science (are preferred to all other existing hypotheses at a certain moment).Here the emprical corroboration is crucial,no matter how ugly or complex a scientific hypothesis is (the standard formalism of QM is rather ugly,for example).

    But when more hypotheses,more or less equally supported empirically,are available the 'aesthetics' part could play a role,sometimes.The definiton of 'beauty' in science depend upon the designated goals of science,or science is intrinsically pragmatic.As Popper (Einstein too) wrote once the first goal of science is to give an account of observed facts based on the minimum number of assumptions,thus we should prefer the hypotheses which proceeds from unifying principles,stable in time,in cases when more,approximatively equal empirically,hypotheses exist (one of the most accepted definitions of 'simplicity' among scientists even now).

    Thus 'beauty' is virtually identified with 'simplicity' (including also the ease to use it for all our practical purposes).Kepler's (an improvment over Copernicus') system is the paradigm here;it is an elegant,simple and stable,hypothesis unlike Ptolemy's,which could basically explain equally well the observed facts (at the time),but with the expense of adding new epicicles and equants,basically on a daily basis.
    Last edited: Jul 18, 2004
  19. Jul 18, 2004 #18
    Besides the “a posteriori” recognized beauty of some theories, such as Kepler’s system, there is the power of beauty in the development of the theories. In this respect, we can remember the platonic substrate of the Kepler’s Mysterium Cosmographicum, a theory later refuted by the observational evidence.
    I think interesting to see the role that Aesthetics plays in the development of Science, in both senses: towards a predictive, simple and consistent theory, or, on the contrary, towards a construct whitout empirical basis. Kepler is a good example of both situations.
  20. Jul 18, 2004 #19
    A theory is beautiful when it can cause any progress....and it is perfect when it can explain everything.
  21. Jul 18, 2004 #20
    Progress isn't synonimous of beauty.
    What is progress? There are advances in discovery and advances in explanation.
    Discovery in many fields of knowledge is facilitated merely by using new methods, without need of new theories. For example, in Biology with the use of the PCR (Polymerase chain reaction), or in Particle Physics with the advances in technology.
    I believe that beauty in a theory is associated, in part at least, to its power of prediction, but not imperiously to progress. Progress can be quantified in some cases. Beauty perhaps never.
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