BEAUTY: The Mathematical Idea of Symmetry
I cannot claim to understand very fully the sense in which mathematicians use the concept of “symmetry.” It does seems to be in some meaningful sense kindred with the concept of symmetry that we use in other, more directly aesthetic contexts– a concept that figures prominently in the human concept of beauty. And so it seems quite possible that whatever is true of the mathematical concept of symmetry would also help to illuminate something meaningful about the nature and meaning of beauty. Just how that “something meaningful” should be understood, however, I don’t yet feel able to articulate.
The following is from the book, THE UNIVERSE AND THE TEACUP: THE MATHEMATICS OF TRUTH AND BEAUTY by K.C. Cole.
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“Many of the most beautiful patterns created by nature and human nature have a great deal of symmetry– tiling patterns and decorative borders and snowflakes and daisies all take a simple pattern and rotate it or flip it or turn it upside down.”
(p. 176)
“The search for symmetry turns out to be a very effective tool for looking beneath superficial differences that camouflage similarities to find a more substantive, permanent meaning. Symmetry therefore lends a satisfying concreteness to the vague sense that there is beauty in truth and truth to beauty. So many of the things people admire are symmetrical: whether they are natural symmetries, like snail shells, or human-made symmetries, like codes of law that attempt to impose equal outcomes on both sides of an argument. [ABS: I'd have used Justice with her scales here to talk about symmetry in "codes of law," rather than this "equal outcomes on both sides of an argument.] It is nice to know that there’s a real quantitative connection between things we admire for aesthetic reasons and things that steer us toward a deep understanding of nature, including, perhaps, human nature.
“Scientists have known about this connection for a long time. Physicist Herman Weyl, who wrote the classic book on symmetry, puts it this way: “My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.” [AS: Is this really putting that same "it" in a different way?]
“Beauty in the mathematical sense is a lot more than a pretty face. It is a way of distilling the essence of things out of the messy mix that nature presents us. Edward Rothstein, trained both as a musician and a mathematician, writes that when we search for symmetries, we are ‘defining which aspects…we find essential and which aspects are irrelevant.’
…”‘What better way to get at the fundamentals of structure than by successive transformations to strip away the secondary properties,’ writes James R. Newman in The World of Mathematics, in a prelude to a section on symmetry.”
(pp. 173-174)
January 28th, 2010 at 10:16 am
Re cast in terms of human experience the said ‘successive transformations’ are known as dis-illusioning experiences and encounters.
Then the remarkable awakening . . .the truth . . . and it was there all the time . . . hidden In Plain Sight.
At first it appears not all that beautiful; no ! But then seeing the over-all scheme of things the beaulty lies in the awesome nature of the seemingly apparent scheme. And There Is No Escape !
January 28th, 2010 at 1:57 pm
Most of the theories scientists postulate are just plain wrong. But they think thay can discover a quality in nature called “super symmetry”. Thus a super collider has been built to determine if just such a quality in nature exists. While mathematicians ignore the secondary properties, the theory of everything will include the symmetrical with the asymmetrical; then everything will be beautiful.
January 28th, 2010 at 2:19 pm
I’m wondering how NSB became so fortunate as to have a regular participant who knows more than the whole community of scientists combined.
January 28th, 2010 at 6:43 pm
A distinguished scientist made that remark. I`d have to go back and try and find it, but this new device of theirs might explain, why scientists are not able to prove their theories. When everyone is blind, it matters not, how many are in that group.
January 28th, 2010 at 6:49 pm
A cosmologist from Arizona State, Lawrence Krauss, pointed this out!
The worrying continued. Lawrence Krauss, a cosmologist from Arizona State, said that most theories were wrong.
http://www.nytimes.com/2010/01/26/science/26essay.html
January 28th, 2010 at 7:26 pm
But Krauss is right, eh, despite being a scientist?
Actually, it depends on what the meaning of “wrong” is. Doesn’t it?
What do humans have in terms of understandings of the world that are less wrong than scientific theories?
January 28th, 2010 at 8:57 pm
It depends on your viewpoint, the angle of sight that enters your view. Is it immoral, unethical, or illegal, contrary to the principles of justice or goodness. But Krauss`s view, from his viewpoint is that, they are mistaken.
Scientific theories are unproven mathematical models, that require confirmation and, scientists have not provided that, that is why Einstein`s theories have been under the scrutiny of validation for all of these many years. i.e. the `Big Bang` is just a theory, but if you don`t accept it, your peers will think you odd.
A scientist cannot save you, if he does not know, but a child can lead you to safety if he does. You can view that moral in `The Posideon Adventure` where the world is turned upside down and noone knows the way out,
The Prophets can see what science cannot understand outside of their precision of calculation.
January 28th, 2010 at 9:25 pm
Tell me, James, do you think that, in science, a theory is the same thing as a hypothesis? Do you think that there is something that a THEORY graduates into, that’s more proven, once it has undergone extensive confirmation?
January 28th, 2010 at 9:34 pm
Andy, I hate to see you expending your time and energy on dealing with James’s mindless pronouncements. Surely you can’t think that the rest of us here on NSB need help seeing his empty pontifications for what they are.
January 28th, 2010 at 9:49 pm
I hav`nt thought about that, but the theory of everything has not been established, `cause gravitation is not understood. These ideas, theories in physics, cannot be confirmed; the equations fall off the page -on and on – and the tools are too primitive to allow progress. The scientists in the field of physics, are worried. I`ll take a crack at it, later. theory …. hypothesis … wish I`d stayed in school. Sir Isaac Newton had no problem with theory, but he was too busy going about the business of the King, to bother much, going further with scientific theory.
January 28th, 2010 at 10:23 pm
Clicked on awaiting the liberal progressives’ observations or comments on Barack Obama addressing the world on the State of the Union.
The Sound of Silence. So be it.
I guess the most interesting thing at the moment is what is hypothesis and what is theory. Without referring to other’s definitions, what do you think maybe
Theory is the best explanation one has for observed behaviors of phenomena
Hypothesis maybe is a notion yet to be tested . . could develop into Theory
depending on, say, if it holds up under scrutiny and challenge.
Of course James is correct in that Theories are only the best we have at the time for many phenomena and over time other observations, or observed changes even, lead to revisions of the best Theories or maybe even abandonment or rejection altogether.
Sometimes it may be more sane to acknowledge that We Just Do Not Know.
(of course that is anathema to the ‘scientific’ mind. That might be the ‘scientists’ fear James comment mentions.)
James often copies some significant writings. I give him extra credit for knowing where to look.
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January 28th, 2010 at 10:26 pm
Theory is the best explanation one has for observed behaviors OR phenomena.
January 28th, 2010 at 10:41 pm
The quotes in the original piece above don’t contain any formulas. Most of my first year studies in architecture at Carnegie-Mellon back in the 60′s involved exploring the various formal principles of symmetry as they can be expressed in 2 and 3 dimensions, and while geometry and mathematics certainly underpinned the work, it all came out one way or another as a visually appreciated expression that could usually be termed “beautiful”. Music shares that as well. The fundamental tonal intervals are all based on the physical divisions of a vibrating string. Halves (octaves), thirds, fourths, fifths, etc. all have a “just right” quality that a sensitive ear naturally picks up. The passing tones are usually arbitrary. I can’t speak about the more esoteric realms of mathematics, but there is a kind of science to be found in much of what we consider to be aesthetically pleasing.
January 28th, 2010 at 10:48 pm
Theory becomes Principle when it can be demonstrably
applied producing consistently observable and/or practical results.
We have applied science all around us today . .
hypothesis . . became theory . . became principle . . proven and applied on and on with consistent results.
January 29th, 2010 at 1:05 am
It is no wonder that political conservatives can get away with their blatant anti-science nonsense. There’s quite an audience for it.
January 29th, 2010 at 8:16 am
Theory, therefore is a form of belief system, that provides the impluse to go forward in the quest for a solution, while hypothesis is an assumption formed as a tentative test to draw out a consequence. So a hypothesis shows that there is insuffucient evidence available to provide more than a tentative argument. Theory, is mere speculation, a form of abstract thought similar to the idea, that all children want to learn. An example of a theoretical concept, fully researched, is Foucault`s theory of human`s impulse, not to help their fellow man, but to control them:
“The prevaqiling theme in Foucault`s philosophy is that human relations are defined by the struggle for power. Right and wrong, truth and falsehood are illusions. They are the creation of language and the will to dominate … Thus, there is no such thing as benevolence: men have created hospitals, schools and priswons not to cure, educate and reform, but to control and dominate`the Other`. The rationalism of the Elightenment was merely a mask for this malevolent impulse. [1]
[1] Patrick West (New Statesman): excerpt – “The Great Philosophers” Jeremy Stangroom, James Garvey ISBN 1-84193-299-X
January 29th, 2010 at 12:48 pm
James, as we talk here about knowledge I am wondering about how much self-knowledge you have about why you express here the things you do. You must know, don’t you, how consistently negative the views you express here are? When some were taken with “Yes we can!” your constant refrain was “No we can’t!” Here on this thread, a discussion of mathematical beauty you took as an opportunity to declare that science basically doesn’t know anything. So, by implication, “No we can’t” becomes no we can’t know anything.
You do see this pattern yourself, don’t you? If you do know that much about yourself, do you also understand why it is that you so consistently take this Nihilistic view of everything? Do you have any insight into what it is in your life that makes it appealing to undercut every belief in anything positive in the human scene? Do you understand what it is that this negativity does for you?
January 29th, 2010 at 8:41 pm
Harry,
In contending with James you are raising a question about your own understanding.
What if we ask you just exactly what per cent of accepted and taught Psychology actually explains the human psyche and the core drives
and instincts of humanity ?
What per cent of accepted and taught ‘scientific understanding’ of the beginnings of the earth, the universe and humanity is truly valid ?
The pretentions to knowledge seem to satisfy the ‘intellectual’ propensities
of a certain set of liberal academics and some so called sciences belong in that field as well.
And the sad case is that as the erroneous notions are reuired subjects sometimes for those in otherwise practical fields of study erroneous understanding drifts on out into the real world until it corrupts social life
as false ‘knowledge’ sends confident ignorami out into the public arena.
How much do you actually KNOW ?
So !
I appreciate James now and then posting it somewhat as it really is
(now and then).
January 29th, 2010 at 9:37 pm
Oh, yes, David R., I can understand how you’d appreciate someone who disrespects science. I’ve been reading your comments for years, you who have this “science background” but are contemptuous of the volumes of research done by cosmologists, because they contradict the account given in Genesis, and likewise use the word “evolutionist” as an epithet of condemnation, because the whole field of biology has failed to confirm the account of life’s creation given in the Bible.
Pretentions? Liberal academics? False knowledge?
Your put-downs are just a way of protecting your cherished beliefs. That’s your right. You can expect sympathy from me about that. But don’t expect me to respect an approach to knowledge that requires rejecting whole fields of scientific knowledge hard won through the best means humans have devised for the search for truth.
January 29th, 2010 at 10:01 pm
It is no wonder that political conservatives can get away with their blatant anti-science nonsense. There’s quite an audience for it.