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Beautiful Ideas.
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Do you find the natural world beautiful?
Does this world embody beautiful ideas?
Does this world embody beautiful ideas?
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jomifl; We may have invented certain methodology in mathematics but we did not invent the embedded mathematics in nature, we, as P. demonstrated, only discovered it. When you look at the world around you the development of man has been largely one of discovery more than invention.
woofgang; In the context of this question, it isn't about perception as a requirement of existence. Yes, the tree may have died, and without observation the fact would be unknown, but the mathematics underpinning nature, as I said above, existed before us and will exist after we have gone.
"Every [good] scientist becomes convinced that the laws of nature manifest the existence of a spirit vastly superior to that of man.......this firm belief in a superior mind that reveals itself in the world of experience, represents my conception of God".
Albert Einstein
woofgang; In the context of this question, it isn't about perception as a requirement of existence. Yes, the tree may have died, and without observation the fact would be unknown, but the mathematics underpinning nature, as I said above, existed before us and will exist after we have gone.
"Every [good] scientist becomes convinced that the laws of nature manifest the existence of a spirit vastly superior to that of man.......this firm belief in a superior mind that reveals itself in the world of experience, represents my conception of God".
Albert Einstein
I was rereading Khandro's post, that was effectively about the irreducible complexity/ high degree of improbability of random emergence of DNA, and it occurred to me that this post essentially then becomes a continuation of that theme, ie "the world is beautiful because it was made so". I'd like to offer an alternative explanation: the world is beautiful because it couldn't really be anything else. Also (expanding on my brief throwaway remark on the last page that wasn't really picked up on) the world isn't that beautiful really anyway.
The key firstly is to be a bit more precise about what *I* mean when I talk about beauty, because there's no real debate to be had if things become too subjective. You say Picasso is beautiful, I say I don't like it one bit (ditto most art -- my lack of interest in pretty much every painting ever is something I'm not really proud of, but never mind). So by beauty here, I mean an innate symmetry of some kind. Colour aside, most flowers have an appealing look because they are roughly symmetrical -- usually circular, rather than square-ish, and with petals that are evenly-spaced and of the same size, so that there is a high degree of approximate symmetry to the structure. (On the other hand, most flowers these days are roughly symmetrical because that's the most appealing look to humans, and deliberate selective breeding plays a role in how many such flowers look these days too. I'll gloss over this as it doesn't really matter.). Long story short, something is "beautiful" if it is in some sense symmetric. Even the relationships between musical notes can fit into this definition: the ratio of two nice-sounding notes in a chord is (almost) entirely independent of their frequency (just to *** off Khandro, the ratio of a note and its perfect fifth on a modern piano is about 1.498 rather than 1.5, ie not exactly 2:3 (sorry, Pythagoras!)). This sort of symmetry would be a "scaling" symmetry, rather than the reflectional or rotational symmetries seen in flowers (although actually some plants have scaling symmetries too, eg the leaves of bracken).
At any rate, there is symmetry of a sort dotted across nature, in most of the places you care to look, and often this symmetry can be associated with beauty in the perception. But it can be associated with something rather a lot more profound, which is to say a certain "efficiency". One of the single most fundamental laws in the Universe is that things are, in some sense, lazy after all (I was not being facetious earlier). Throw a ball in the air, and under the influence of gravity it will form a smoothly curved path. Why? Because, you can say, a jagged one would have been rather a lot less efficient. The same idea would explain why musical notes are sinusoidal rather than anything else -- because it's the smoothest way of transmitting (sound) energy. This principle is known as [Hamilton's] Stationary Action Principle, and it's vital reading for anyone who is going to try and dabble in the philosophy of science and nature.
The two concepts, of symmetry and economy, are linked, by one of the most wonderful pieces of mathematics ever, called Noether's Theorem (another vital thing to know about), which could be taken as saying that the more symmetric a system is, the stabler (and so more economical) it is over time. Conversely, the fewer symmetries a system has, the more it has a tendency to evolve away from any kind of order or stability.
Thus it is that, say, the Earth is (roughly) spherical, because a cubical planet is unstable under rotation and the effects of gravity; while a planet that didn't rotate would end up developing horrible temperature gradients that disrupted any attempt life made to get going. Thus, also, the symmetry in flowers, because it would certainly end up taking a great deal more effort and energy creating an arbitrary asymmetric design.
Therefore, symmetry is not here to brighten our existence, but to enable it.
TBC...
The key firstly is to be a bit more precise about what *I* mean when I talk about beauty, because there's no real debate to be had if things become too subjective. You say Picasso is beautiful, I say I don't like it one bit (ditto most art -- my lack of interest in pretty much every painting ever is something I'm not really proud of, but never mind). So by beauty here, I mean an innate symmetry of some kind. Colour aside, most flowers have an appealing look because they are roughly symmetrical -- usually circular, rather than square-ish, and with petals that are evenly-spaced and of the same size, so that there is a high degree of approximate symmetry to the structure. (On the other hand, most flowers these days are roughly symmetrical because that's the most appealing look to humans, and deliberate selective breeding plays a role in how many such flowers look these days too. I'll gloss over this as it doesn't really matter.). Long story short, something is "beautiful" if it is in some sense symmetric. Even the relationships between musical notes can fit into this definition: the ratio of two nice-sounding notes in a chord is (almost) entirely independent of their frequency (just to *** off Khandro, the ratio of a note and its perfect fifth on a modern piano is about 1.498 rather than 1.5, ie not exactly 2:3 (sorry, Pythagoras!)). This sort of symmetry would be a "scaling" symmetry, rather than the reflectional or rotational symmetries seen in flowers (although actually some plants have scaling symmetries too, eg the leaves of bracken).
At any rate, there is symmetry of a sort dotted across nature, in most of the places you care to look, and often this symmetry can be associated with beauty in the perception. But it can be associated with something rather a lot more profound, which is to say a certain "efficiency". One of the single most fundamental laws in the Universe is that things are, in some sense, lazy after all (I was not being facetious earlier). Throw a ball in the air, and under the influence of gravity it will form a smoothly curved path. Why? Because, you can say, a jagged one would have been rather a lot less efficient. The same idea would explain why musical notes are sinusoidal rather than anything else -- because it's the smoothest way of transmitting (sound) energy. This principle is known as [Hamilton's] Stationary Action Principle, and it's vital reading for anyone who is going to try and dabble in the philosophy of science and nature.
The two concepts, of symmetry and economy, are linked, by one of the most wonderful pieces of mathematics ever, called Noether's Theorem (another vital thing to know about), which could be taken as saying that the more symmetric a system is, the stabler (and so more economical) it is over time. Conversely, the fewer symmetries a system has, the more it has a tendency to evolve away from any kind of order or stability.
Thus it is that, say, the Earth is (roughly) spherical, because a cubical planet is unstable under rotation and the effects of gravity; while a planet that didn't rotate would end up developing horrible temperature gradients that disrupted any attempt life made to get going. Thus, also, the symmetry in flowers, because it would certainly end up taking a great deal more effort and energy creating an arbitrary asymmetric design.
Therefore, symmetry is not here to brighten our existence, but to enable it.
TBC...
On the other hand, too much symmetry is actually a bad thing, and just as symmetries are important it is the imperfections that drive real change and progress. As a mundane example I refer to the ratio of fifths earlier, not 2:3 but 2.0022:3. Also, you might note that faces are rarely if ever perfectly symmetric; while the most beautiful faces are probably fairly close, equally it's important that they aren't exactly reflections one half of the other, and as you snap into perfect symmetry faces tend to look, if not always awful, then certainly a bit off. It is the subtle imperfections that differentiate between beauty, or at least humanity, and some sort of artificial construction that can just look awful.
At a more fundamental level, the asymmetries or broken (ie almost-but-not-quite) symmetries turn out to be vital for getting everything rolling. As a physicist I'm going to quote a set of physics examples: I'd recommend reading further into these, if you get the chance (and aren't too exhausted by reading these already quite lengthy posts).
The first example is the asymmetry between matter and antimatter, still not completely understood. It the two were perfectly symmetric version of each other, then it follows that if you start with, say, equal quantities of matter and antimatter in the universe, then after a certain amount of time there would still be equal quantities of matter and antimatter, meeting and annihilating, and just generally messing up much hope of things just being stable. But we don't see this, and for whatever reason matter/ antimatter isn't a perfect symmetry. But because of that, here we are to moan about it.
As a second example, the Higgs boson is really just a manifestation of another symmetry that we should all be bloody grateful isn't perfect after all. This is the "electroweak" symmetry, that holds that two of the four currently known forces are really the same. Except that, if they were *exactly* the same, then nothing could have a mass (for various technical reasons, including Noether's theorem again). We have a mass, but this theory also works stupidly well at describing things. To reconcile the two is one of the more insane pieces of mathematical physics -- and, arguably, we still haven't reconciled it properly yet -- but it is doable and ends up implying both the importance of a symmetry and the occasional necessity for destroying that "perfection".
The third example I will give is in the currently completely theoretical concept of a Grand Unified Theory (GUT). These take various forms and, so far, they are all wrong. You might wonder, then, what motivated physicists to try them, then, and the answer, ironically enough, is an obsession with imposing as much symmetry as possible.
At this point in a typical particle physics lecture, your speaker would moan a lot about the unsolved problems of physics, and how "ugly" the Standard Model is, and bla bla bla. The GUT approach is to take the already fairly tightly-packed symmetries present, squeeze them into an consolidated symmetry, and then voila everything is perfect at last! Or not. In this case, the problem is that unifying symmetries has the unfortunate consequence of mixing things together that are observed to be separate. In particular, a proton can decay into an electron (+other)), but protons are stable particles with an observed lifetime of approximately 100 million trillion times longer than the age of the Universe, and the only way to fix this problem is to basically screw around with the theory until it stops being actually useful. Too much symmetry, then, is a bad thing just as much as too little symmetry can be.
* * *
So in summary, then (I really don't want a third post), beauty, in the form of symmetry, is necessary for existence (not ours, per se, but of anything) but too much symmetry is bad for you. It's too constraining. Sometimes, it is the ugly stuff that makes the world go round.
At a more fundamental level, the asymmetries or broken (ie almost-but-not-quite) symmetries turn out to be vital for getting everything rolling. As a physicist I'm going to quote a set of physics examples: I'd recommend reading further into these, if you get the chance (and aren't too exhausted by reading these already quite lengthy posts).
The first example is the asymmetry between matter and antimatter, still not completely understood. It the two were perfectly symmetric version of each other, then it follows that if you start with, say, equal quantities of matter and antimatter in the universe, then after a certain amount of time there would still be equal quantities of matter and antimatter, meeting and annihilating, and just generally messing up much hope of things just being stable. But we don't see this, and for whatever reason matter/ antimatter isn't a perfect symmetry. But because of that, here we are to moan about it.
As a second example, the Higgs boson is really just a manifestation of another symmetry that we should all be bloody grateful isn't perfect after all. This is the "electroweak" symmetry, that holds that two of the four currently known forces are really the same. Except that, if they were *exactly* the same, then nothing could have a mass (for various technical reasons, including Noether's theorem again). We have a mass, but this theory also works stupidly well at describing things. To reconcile the two is one of the more insane pieces of mathematical physics -- and, arguably, we still haven't reconciled it properly yet -- but it is doable and ends up implying both the importance of a symmetry and the occasional necessity for destroying that "perfection".
The third example I will give is in the currently completely theoretical concept of a Grand Unified Theory (GUT). These take various forms and, so far, they are all wrong. You might wonder, then, what motivated physicists to try them, then, and the answer, ironically enough, is an obsession with imposing as much symmetry as possible.
At this point in a typical particle physics lecture, your speaker would moan a lot about the unsolved problems of physics, and how "ugly" the Standard Model is, and bla bla bla. The GUT approach is to take the already fairly tightly-packed symmetries present, squeeze them into an consolidated symmetry, and then voila everything is perfect at last! Or not. In this case, the problem is that unifying symmetries has the unfortunate consequence of mixing things together that are observed to be separate. In particular, a proton can decay into an electron (+other)), but protons are stable particles with an observed lifetime of approximately 100 million trillion times longer than the age of the Universe, and the only way to fix this problem is to basically screw around with the theory until it stops being actually useful. Too much symmetry, then, is a bad thing just as much as too little symmetry can be.
* * *
So in summary, then (I really don't want a third post), beauty, in the form of symmetry, is necessary for existence (not ours, per se, but of anything) but too much symmetry is bad for you. It's too constraining. Sometimes, it is the ugly stuff that makes the world go round.
Khandro,/the embedded mathematics in nature/.
So when a bee lands on a flower you are say that it carries out mathematical computations in order to facilitate the activity or even to make it possible. When I drop a stone it calculates the rate at which it has to accelerate as it already knows the local gravity constant? The English language existed before the English people..it being a language like mathematics..
I think you need to learn to distinguish between the message and the messenger.
So when a bee lands on a flower you are say that it carries out mathematical computations in order to facilitate the activity or even to make it possible. When I drop a stone it calculates the rate at which it has to accelerate as it already knows the local gravity constant? The English language existed before the English people..it being a language like mathematics..
I think you need to learn to distinguish between the message and the messenger.
jim; Great contribution, but symmetry in nature is never perfect I think you labour that point too much, one side of a tree isn't the mirror image of the other side, there is though within the structure a search for balance. It is in fact, the lack of perfection that gives the stateliness and symmetry to the tree.
similarly, // the ratio of a note and its perfect fifth on a modern piano is about 1.498 rather than 1.5, ie not exactly 2:3//
If the cousin of a friend of mine who prepares concert pianos read that, I think she would laugh. The process which a musical note goes through before it registers in a brain is quite involved, I could spell it out but it would take a while. There's no need to apologise to Pythagoras, the mathematic principle remains intact.
The art and architecture of humans has nothing to do with it at all, some art attempts to be deliberately ugly (German expressionism for example) and the wonders of Baroque architecture are based entirely on a non-symmetrical aesthetic.
more later.
similarly, // the ratio of a note and its perfect fifth on a modern piano is about 1.498 rather than 1.5, ie not exactly 2:3//
If the cousin of a friend of mine who prepares concert pianos read that, I think she would laugh. The process which a musical note goes through before it registers in a brain is quite involved, I could spell it out but it would take a while. There's no need to apologise to Pythagoras, the mathematic principle remains intact.
The art and architecture of humans has nothing to do with it at all, some art attempts to be deliberately ugly (German expressionism for example) and the wonders of Baroque architecture are based entirely on a non-symmetrical aesthetic.
more later.
"If the cousin of a friend of mine who prepares concert pianos read that, I think she would laugh."
Well, I'd be surprised by that since I went and looked up the tuning table for pianos, eg:
http://
http://
http://
and then went and checked the ratios. You can do it for yourself if you like: the answer varies (another key point!) but is always close to 1.498 rather than 1.5 for the G/C-like ratio. It was mainly a point of interest than anything else. What with hearing involved I daresay the ear will end up modifying things a little again, perhaps restoring an effective 2:3 ratio for all I care, but I doubt it. Anyway, the ratio on a modern piano at least is not 2:3 (although it's still as near as dammit, which reinforces the point I was making).
I'm also a bit intrigued by the "it is, in fact, the lack of perfection..." point, that almost looks as if it's meant to be arguing with my post while instead managing to repeat it. But never mind. I did go on a bit.
Art, though -- well, I started off by deliberately sidestepping that, because I confess that much of the painted medium of art I find fairly tedious, really. As I said earlier I'm almost embarrassed by this, but it is true, and I tend to find only things in motion interesting, ie music and film; in that context, we can't really have a debate about the origin of beauty in art because we'll never agree on what in art is beautiful.
Well, I'd be surprised by that since I went and looked up the tuning table for pianos, eg:
http://
http://
http://
and then went and checked the ratios. You can do it for yourself if you like: the answer varies (another key point!) but is always close to 1.498 rather than 1.5 for the G/C-like ratio. It was mainly a point of interest than anything else. What with hearing involved I daresay the ear will end up modifying things a little again, perhaps restoring an effective 2:3 ratio for all I care, but I doubt it. Anyway, the ratio on a modern piano at least is not 2:3 (although it's still as near as dammit, which reinforces the point I was making).
I'm also a bit intrigued by the "it is, in fact, the lack of perfection..." point, that almost looks as if it's meant to be arguing with my post while instead managing to repeat it. But never mind. I did go on a bit.
Art, though -- well, I started off by deliberately sidestepping that, because I confess that much of the painted medium of art I find fairly tedious, really. As I said earlier I'm almost embarrassed by this, but it is true, and I tend to find only things in motion interesting, ie music and film; in that context, we can't really have a debate about the origin of beauty in art because we'll never agree on what in art is beautiful.
jim360//the ratio of a note and its perfect fifth on a modern piano is about 1.498 rather than 1.5, ie not exactly 2:3 //
Incorrect. The Perfect Fifth is 2:3 and always will be.
The 1.498 ratio in modern instruments is the result of Equal Temperament which compromises the exact ratios as described by Pythagoras in order to allow the instrument to be played in any key without retuning.
It is one of many such systems which deviated from perfect tuning but has come to virtually completely dominate modern music. As such our perceptions of pitch are so accustomed to it that perfect tuning may sound odd to many ears.
However, guitarists will sometimes "bend" the fifth by stretching the string, lifting its pitch to the perfect fifth.
Incorrect. The Perfect Fifth is 2:3 and always will be.
The 1.498 ratio in modern instruments is the result of Equal Temperament which compromises the exact ratios as described by Pythagoras in order to allow the instrument to be played in any key without retuning.
It is one of many such systems which deviated from perfect tuning but has come to virtually completely dominate modern music. As such our perceptions of pitch are so accustomed to it that perfect tuning may sound odd to many ears.
However, guitarists will sometimes "bend" the fifth by stretching the string, lifting its pitch to the perfect fifth.
beso; Thank you!
jim; The vibration of a string goes through several transformations before arriving at our minds as a message. The vibration disturbs the surrounding air directly, simply by pushing it. The hum of an isolated string is quite weak however. Practical musical instruments employ sounding boards which respond to the string's vibration with vibrations of their own. The motion of the sounding board pushes the sound around more robustly.
The disturbance of air near the string or surrounding board then takes on a life of its own, becoming a propagating disturbance: a sound wave that spreads outwards in all directions. Any sound wave is a recurring cycle of compression and decompression. The vibrating air in each region of space exerts pressure on neighbouring regions and sets them into vibration.
Eventually, a portion of this sound wave, funneled by the complicated geometry of the ear arrives at a membrane called the eardrum a few centimeters within. Our eardrums serve as inverse sounding boards, where now vibrations of air induce mechanical motion, instead of the opposite.
We have now arrived at the eardrum, and I could continue ad nauseam about how it reaches the brain via ossicles, cochlear fluid, basilar membranes and hair cells, but enough! Suffice to say that Pythagoras's rules, translated into frequency, show that notes sound good together if their frequencies are in ratios of small whole numbers.
jim; The vibration of a string goes through several transformations before arriving at our minds as a message. The vibration disturbs the surrounding air directly, simply by pushing it. The hum of an isolated string is quite weak however. Practical musical instruments employ sounding boards which respond to the string's vibration with vibrations of their own. The motion of the sounding board pushes the sound around more robustly.
The disturbance of air near the string or surrounding board then takes on a life of its own, becoming a propagating disturbance: a sound wave that spreads outwards in all directions. Any sound wave is a recurring cycle of compression and decompression. The vibrating air in each region of space exerts pressure on neighbouring regions and sets them into vibration.
Eventually, a portion of this sound wave, funneled by the complicated geometry of the ear arrives at a membrane called the eardrum a few centimeters within. Our eardrums serve as inverse sounding boards, where now vibrations of air induce mechanical motion, instead of the opposite.
We have now arrived at the eardrum, and I could continue ad nauseam about how it reaches the brain via ossicles, cochlear fluid, basilar membranes and hair cells, but enough! Suffice to say that Pythagoras's rules, translated into frequency, show that notes sound good together if their frequencies are in ratios of small whole numbers.
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