The World of Mathematics is a four-volume set first published in 1956 under the aegis of editor James Newman. In the fourth volume there is a thirteen-page piece which purports to be "A Mathematical Theory of Art," by George David Birkhoff, a math prof who worked at Princeton University.
Birkhoff's work on a mathematical measurement of beauty was first given a public forum in Italy in 1928, and later published in book form (Aesthetic Measure, Cambridge, 1933). The article that appears in Newman's volume, is a shorter piece that was first published in Italy in 1928.
Birkhoff argues that although aesthetics is a matter of intuition and that it is singular, we can nevertheless find general principles that can be reduced to mathematical regularity.
Here is the general theory of aesthetics that Birkhoff proposes:
Aesthetic merit equals complexity divided by order.
M = O/C, in its shortest form, in which M stands for Aesthetic Merit, and O equals order, and C equals complexity.
In general, this formulation seems to derive from Thomas Aquinas' formula (without attribution) which says that beauty is a complex coherence, or a coherent complexity, in which a singularity pervades multiplicity. Examples include the Rose Window at Notre Dame Cathedral, which offers many scenes from the Bible unified by a circular window frame, and a common color scheme.
This formula is one of the ways in which I judge student essays: if there is a strong driving argument that takes many disparate facts and frames them into a new and convincing order, the student stands likely of receiving an A.
But the formula is of course not enough!
A musical composition ranges through a theme that is announced in the first note, and generally returns to that same note at the end, which provides the time-sequence with unity. But we can't just hum the same bar twenty times, as there would not be enough complexity. Birkhoff writes, "...a poem overburdened with alliteration and assonance fatigues by undue repetition..." (2191) [perhaps he was thinking of Charles Swinburne]. "...a musical performance in which a single wrong note is heard is marred by the unnecessary imperfection..." (2191).
Birkhoff admits that art is not purely formal. It means something beyond itself. A perfectly beautiful watercolor, if it happens to be by Adolf Hitler, throws an ugly shadow by dint of what else it means. Birkhoff writes, "In fact the symbols O and C represent social values, and share in the uncertainty common to such values" (2194).
Birkhoff's formulation nevertheless cannot give us a formula for the creation of beauty. It can only help us to understand "after the fact" why something is beautiful. Birkhoff argues that it is the "consensus of opinion" for those who are familiar with some art form, which determines the canon of what is beautiful in that arena. In dog shows, it's dog experts. We may have a quorum therefore of sixteen experts who agree that one specific example of a Scottish Terrier is the most beautiful. However, little Billy next door who has a mangy Scottish Terrier named Sparky, may not agree with the quorum. His unsightly little mutt will no doubt be much more beautiful to him, because of the love with which he regards his (to us) comical cur.
Symmetry is not sufficient to establish beauty. There is also connotation. Formal or geometrical elements of a painting or a dog are one thing in establishing the beautiful. But context is another. If the canine has its teeth lodged in your ankle, or in your baby's ear, it is likely to be regarded with less disinterested contemplation, with less of a view to its formal arrangement, and more of a view to kicking its behind.
"The feeling of 'empathy,' whose importance has been stressed by the psychologist Lipps, contributes to the enhancement of the aesthetic effect. Similarly, actual participation on the part of the percipient, as in the case of singing a tune as well as hearing it, will enhance the effect" 2188). [Singing church hymns definitely does enhance their effect, so this part we may agree with.]
In a church hymn or a piece of music in general, "complexity C will be measured by the number of notes in the melody" (2188). Birkhoff does not describe order so neatly, but I think in this case it would be the structure of the notes, or their composition. The composition should be such that we can sense the overall order (that there IS an order, and that the order is pleasingly complex, and yet still coherent enough that we can recognize that the whole is of a piece).
Look at the following YouTube video with Katy Perry singing Hot and Cold, for a very contemporary example of a song that rises through a series of choruses to make a single unified statement out of a complexity of material:
http://www.youtube.com/watch?v=X75mry1LcFg
Katy Perry is a contemporary singer (my daughter who's nine wanted me to see this video as she wants to be a singer, something I'm hoping will pass!). She released a Christian album when she was sixteen (her parents were evangelicals), and has apparently now gone in a more secular direction. Notes of unity: the drumming, the wedding itself, the shoulders bouncing (including that of the little girl's on the right, indicating a certain sisterhood of the females present, the chorus about breaking up (one into many), with the reversal also vying for position, in which the young man and woman move from two (complexity) to one, through the marriage vow, a promise of eternity together, and the men celebrating this, too. When many things form into a pleasing new combination, it's beautiful. That's more or less what Birkhoff is telling us about beauty. But I think his formula could be simplified..
E pluribus unum, might be another way of looking at it. When many things equal one, by virtue of their inner coherence, there is beauty to be found. Another simpler formula, might be
X = 1.
Where X equals anything more than 1, and where the greater the number of original things that go into X still equals one, the more beautiful it is. Looking up at the stars at night (there are trillions), we name them under the appellation, "Stars." This singular term for a multiplicity is the general drift of Birkhoff's idea. Using Occam's razor, I think my formula is simpler and thus more beautiful than his, but I stand on the shoulders of a giant.
19 comments:
I see that you have been reading (sort-of) your Badiou (Being and Event) and that, perhaps you "balled" before reading/absorbing "Meditation Thirty-Three" ?
then to f u l l y comprehend "the proposition
'mathematics is ontology'" (...) and all of his various and sundried de:notations is far beyond the scope of your post (?)
"bailed" not "balled" ... we gotta protect the children!
sorry for the "slip"
Katy Perry has certainly "gone secular" with her first hit, "I Kissed a Girl"
And that music is hardly beautiful. Boring is more like it.
Also, if she wants to be a singer, I recommend dancing lessons and a vocal processor. Then she's in like Flynn.
I just hope someone will help me with the actual formula in mathematical terms. At first I thought I improved on Birkhoff when I simply wrote that X = 1, but then I think I need to write something about when X > 1, and yet equals one, it RESULTS in beauty, which might suddenly be more cumbersome than Birkhoff's original formulation.
Perhaps George Grady can help.
I also have a line out to two other mathematicians, and will probably extend that to yet a fourth (our provost is a mathematician, and he's almost retired, and has helped me in the past!).
But he too might not like Katy Perry's music. I think the chorus is simply magnificent: breathtaking.
I would also like to say that when X is further from the Zero line toward infinity and YET is still equal to one, that the more beauty there is to contemplate. That Zen formula (all is one), is kind of where I need to head, but I would like to indicate a gradation from simple to complex, with complexity being the more valued. Is that something that can be translated into simple mathematical terms?
G.M. I just googled I Kissed a Girl, and thought the music in that song was very poor by comparison. I mean, it was VERY POOR. What's missing?
What is poor about it?
It's so one-track, for one thing. There is also not nearly as much humor in the visuals -- it's more lurid, and meant merely to titillate.
But the song itself doesn't have enough complexity. It's the same thing drilled in over and over.
What's neat about Hot and Cold is the bringing together of some many opposites -- hot and cold, black and white, (this is also demonstrated visually), and the slow set up, followed by the relatively rapid chorus, and the notion of chaos about to break out, followed by a steady resumption of order with the women hectoring, and the men refusing solidarity with the recalcitrant bridegroom, until he caves, in which case there's general jubilation.
There is also the notion that she's boring, and he wants more complexity, but the inset scene in which she's got sunglasses and says HE'S boring, reverses that angle, too.
It's a stunning little bit of vulgarity, no doubt, but quite charming, no less.
Compared to I Kissed A Girl, which is just glaring vulgarity with no redeeming value, or not enough opposites to bring into one-ness, Hot and cold is much the better song. Would you at least agree with this?
Also, in I Kissed A Girl, she doesn't even know the girl's name, which is ethically wrong, as there's no relation to one another.
In the other song, Hot and Cold, there's a longer relationship. Which is more ethical and friendlier.
It isn't just lust, in other words.
It's a more complex phenomenon (love & marriage), so it's more aesthetically interesting.
Plus, the church is involved in the newer song, as are families, and the whole society, as opposed to it just being a secretive fling inside of anonymity (much simpler and not as appealing to the overall order that eventuates -- like hot air escaping as opposed to being turned into a funnel, or a curlicue, of passion and love intertwined).
Kirby,
I don't understand your various formulas. You state Birkhoff's proposal as:
Aesthetic merit equals complexity divided by order
or M=C/O. How are you defining and measuring complexity and order? Without some indication of this, it just seems like so much mumbo-jumbo. What is complexity? What is order?
I don't think that your examples are really adequate here. Some (such as Birkhoff's conflation of complexity of organ music with length) strike me as just, well, absurdly unthinking.
As for your "X=1", what is X supposed to be? I guess I really don't understand what you're trying to say.
You like the bringing together of opposites?
How's about the Irish baritone national champion of 2003 and a hippie-ish slacker from Austin playing a country song written by an obscure cowboy at an upscale lodge in the Colorado mountains.
http://tinyurl.com/974g6e
The formula is m=c/o -- that is, complexity divided by order.
I believe that Kirby is saying ideally that m=1.
m>1 would result from more complexity and less order.
m<1 would result from less complexity and more order.
One certainly wants to strike a balance between the two.
I'll try to fill in more about what Birkhoff thought constituted complexity and order. It's a fairly vague article, but it's about 13 pages long, and I might be able to fill in these values a little more thoroughly, but I won't be able to get to this until late this afternoon.
GM, in the book, Birkhoff writes the formula like that M = O/C, and writes it out like this:
"Within each class of aesthetic objects, to define the order O and the complexity C so that their ratio M = O/C yields the aesthetic measure of any object of the class" (2186).
"It will be our chief aim to consider various simple classes of aesthetic objects, and in these cases to solve as best we can the fundamental aesthetic problem in the mathematical form just stated. Prelimainry to such actual application, however, it is desirable to indicate the psychological basis of the formula and the conditions under which it can be applied" (2186).
I have the feeling that mathematics is base duse on tiny empirical problems such as the amount of change perceived in a bubble under different pressures. Even that may be too big. But when we're getting to enormous issues like, what is the beautiful, I think that math is just as bewildered as philosophy or any other discipline.
Birkhoff also has a fascinating attempt to quantify ethics mathematically (in the same volume). But the values he assigns are seemingly arbitrary. In international relations for instance he gives an enormously high value to friendship between nations. We shouldn't attack a nation if it wrecks our friendship with them.
Or we should always keep a confidence, since that would wreck a friendship if we did not.
But this creates weird problems. For instance, let's say that I'm the friend of A and B. A says that he's stolen something from B, and asks me to keep this in confidence. If I am truly a friend to A, then I would keep this in confidence. But if I was truly a friend of B, I would tell them? So what do I do?
I know what I would do. I would tell A to give it back, or else I would no longer be their friend, and I would place a further condition. You have to tell B what you did, and you have to apologize, and you have to promise never to do anything of that kind again, or I will no longer be speaking to you.
Depending on what's been stolen, of course. If it's only a pencil, and A is just kidding around by having stolen it, and it hasn't done any real damage to B (who is a pencil maven and has way too many pencils as it is), I might see this in context as ok.
But if A has stolen B's prize camera, and B is distraught, then I would definitely proceed with my initial plan, which would be to oblige a return of the object, with a full apology, and also a promise that nothing of the kind ever take place again.
I keep thinking that math should deal with larger problems than simple quantification. It should also deal with qualities, and with qualification. It may not be able to do this, but it should allow us to think about good and bad, right and wrong, beautiful and ugly.
If it doesn't, what good is it to the humanities?
Well, you said complexity divided by order, which would be c/o.
I would think that complexity would be the primary notion in what is beautiful -- structured on order. That is, the order is within the complexity -- therefore the order is what divides the complexity.
According to the formula, beautiful things approach a 1:1 complex:order ratio.
Now, about your other points, quality is undefinable.
For George, I don't think Birkhoff is every very detailed about how he uses his own formula and he indicates that his theory is only a toe in the water of a vast pool of un-mathematized thinking.
He does say in terms of his formula, that it is very similar to other attempts to quantify beauty.
"The well-known aesthetic demand for 'unity in variety' is evidently closely connected with this formula. The definition of the beautiful as that which gives us the greatest number of ideas in the shortest space of time (formulated by Hemsterhuis in the eighteenth century) is of an analogous nature" (2186).
I think Birkhoff here is confusing the idea. Unity means oneness, so unity in variety would mean, many things, but one major theme, for example. Unity means one-ness.
This is why I think the formula should go,
Beauty is X in which X > 1 = 1.
I don't think this is going to pass the mustard in a mathematics consortium in which absolute precision is termed definitively kosher, and anything that isn't has to be tossed out.
But I think to attempt to create a mathematical formula for 'unity in variety' unity at least should be given the value of 1.
What else is unity if not one-ness?
And it's oneness throughout the new Katy Perry video that is so striking. Unity in variety.
Birkhoff writes along these lines, "A painting should have ONE predominant center of interest upon which the eye can rest" (2191).
At any rate, he does have a longer book called Aesthetic Measure, but right now at Amazon.com the only copies of that book are going for well over a hundred dollars, so I don't have access to it. Perhaps in that book he does give us more concrete descriptions, but in this one chapter (which is apparently culled from the book) he doesn't.
I could probably get the book through interlibrary loan. I will attempt that, and get back to you all on this over the summer after I have had time to read it.
But let's just say that ORDER over COMPLEXITY, finding a unique and single thread in a myriad of surprising details, is the aesthetic experience.
It would be similar to the pleasure of an analogy, in which we find a way of relating two things not generally found to be related.
"Like the chance encounter of an umbrella and a sewing machine on a cadaver table," is the poetic image written by Lautreamont that Breton and Soupault used as the touchstone of their own poetics. In which two unlike things are found to be in some way alike, or to have more to do with one another than one thought.
"A flea like a speck of a tobacco" is a line from Jules Renard.
To see two things as alike, and therefore one.
But perhaps ephemerally one, because this moment also asks us to ask how a speck of tobacco and a flea are also different, so it asks us to think comparatively.
Is it one, is it two? One, two?
Marriage might be like that: are we one, or two?
The aesthetic moment makes the claim that two (or more) disparate things are in fact one.
Kirby, it's Christy from International Arts Movement here... I want to send you an email but can't find your contact info. Can you email me at christy(at)iamny.org? It's about Mako Fujimura...
Kirby,
If I were to try to find a mathematical model for beauty (which I doubt that I will anytime soon, as I suspect it would be a Sisyphean task), I would try to follow a path similar to that followed by mathematicians trying to understand randomness. Namely, what are the important properties that I think beauty should satisfy, and can I find something that models that. That is, how does beauty behave? If I can't handle that, then I have no chance at finding a model for it, and I have no way to evaluate any models I might come up with. If I don't know what beauty "does", I can't tell if my model matches or not. Unfortunately, beauty is much more complicated that randomness and probability, and it took some brilliant mathematicians centuries of thought before Kolmogorov finally found a reasonable mathematization of probability.
George, I had to think about this quite a bit.
Is beauty stable, like a property, a noun, or is it something like a process, a verb?
If you think of an outfielder running to catch a fly ball over his shoulder and jumping up over the home run fence, and catching it just in the top of his glove's webbing, then whap! Beautiful catch.
But the Vietnam Memorial is also beautiful, and doesn't move, although it does MOVE US, but I don't know if that's something we can say -- in general -- that to be moved emotionally is beautiful.
We can be moved by someone's death, but I'm not sure it's always beautiful, especially if it happens in a terrible way. The decapitation of a female student at Virginia Tech a few days ago (I read about it in yesterday's paper) probably moved her friends and family, but I doubt if any of them thought it was beautiful.
It's very hard to define beauty, as you say. But I can't even tell you if it's a static property, or if it's a process, like random-ness.
But I really want to send you a poem I'm working on by the poet Gregory Corso and get your opinion about it. He talks about the history of mathematics in it -- especially Luca Pacioli -- a geometrician who knew Leonardo da Vinci. I want to send it to you via email, because I don't want to share my thinking about it, since I might want to publish my results some day.
If you wouldn't mind sending me your email, I would like to show this to you, and to tell you some other things, too, about it.
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