Sunday, March 01, 2009

Traversing an N-Dimensional Aesthetic Space

I've been musing lately about a problem that has stubbornly resisted my attempts at solving it for quite some time; indeed, I sometimes wonder if I've made any headway at all in all the years I've spent reflecting on it. Perhaps I should pay heed to the title of my own blog - namely tao - and plumb a bit of tao's timeless wisdom. To wit, maybe I ought to treat my problem not as a "thing" that needs solving, but as a transient stepping-stone on a timeless path toward gradual self-enlightenment.

"What is beauty?" I [S Nachmanovitch] asked him that night. He [Gregory Bateson] said, "Seeing the pattern which connects." (quoted from Old Men Ought to be Explorers, by S. Nachmanovitch)

My "problem" is to find the "optimal feature space" in which to describe the aesthetic sensibilities of particular artists; that is, essentially, to find an objective language (or, least as objective a language as possible) to describe the subjective propensities of, and differences between, individual painters, musicians, or photographers. We all "know" the difference between, say, Mozart's music and that of Beethoven; or the difference between a painting by Matisse and another by Picasso. Sometimes the differences, as in these "obvious" cases, are striking. In other cases, the differences may not be so clear cut: if one was, a priori, unfamilar with the works of Minor White and Brett Weston, for example, some of their respective abstracts may appear - superficially at least - as aesthetically indistinguishable.

Somehow, perhaps in the way that Malcolm Gladwell calls "thin slicing" in his book Blink, we all make quick, largely unconscious, assessments about makes one work different from, or similar to, another. We can sometimes analyze - after the fact - why we made the decision of similarity or difference that we made. But (as Gladwell also points out in his book), we are not always able to articulate the precise feature-space decomposition we used to make our rapid-fire decision (because our subconscious thought-process does not always percolate up to the conscious level); nor can we really be sure that whatever feature-space decomposition we are able to articulate is an accurate reflection of what our unconscious information processing. Of course, often our thin-slicing attempts are also simply wrong.

The larger question, even if only as a thought experiment, remains. Let's start small, and not yet all-encompassing - a bit later I will generalize the question from photography to all forms of creative expression - and confine our analysis to photography alone, as an exemplar of a broader class of "art" and its associated larger class of aesthetic possibilities. We ask: what is the optimal set of "features" (to be defined shortly) of "photographs" such that - in the N-dimensional abstract aesthetic space defined by these features as (roughly) orthogonal axes - two conditions are simultaneously satisfied: (1) the differences among photographs is maximized (with respect to sets of photographs produced by individual photographers), and (2) the differences between photographs produced by the same photographer (i.e., between any two images within a given photographer's own oeuvre) are minimized? In a sense, I want to perform a "simple" exercise of mathematical pattern recognition, but without any (or little) initial sense of what space I'm performing it in, or even what I'm setting out to "recognize."

What do I mean by features? Well, any reasonably well-defined "parameter" that can be used to describe a photograph (which may, implicitly, involve both its physical attributes, as a print, and nonphysical attributes, such as subject matter or other contextual primitives). Of course, many different features exist (indeed, the set of possibilities is enormous); but not all features are as important in describing a work as others. More precisely, different sets of features will be better, or worse, at simultaneously identifying the works that are produced by a given photographer and distinguishing among bodies of works produced by different photographers.

Thought Experiment #1. Schematically, we can imagine a 3-dimensional space (in general, the dimension D can be very large) consisting of the features f1, f2 and f3. As a thought experiment, imagine we have the collected works of three of photographers (A, B, and C; that we "code" using the colors red, blue, and green). We classify each photograph, of each photographer, according to where in the feature space in lives. It does not matter whether the "points" in this space are cleanly defined or not; the only thing that matters for this thought experiment is the fact that every work by each of the three photographers is classified according to the values of the three features we have used to define this particular "aesthetic space" F = {f1, f2, f3}. As a concrete example, the three features might be: f1=average hue, f2=degree of local constrast, and f3=number of triangular shapes. And, indeed, as we might expect of such a loose (random almost) set of parameters, we would not be surprised to learn (if we actually went to the trouble of performing this experiment) that these features do little to distinguish among our three photographers. Our plot of their respective oeuvres might look something like this...

But now, suppose we are a bit more clever than this. Suppose, after carefully studying the works of these three photographers, we discover a new set of features - {f1', f2', and f3'} - such that, in this new aesthetic space, F', the same body of work now appears considerably more tightly clustered:

Here we see - by direct visual inspection - an "obvious" distinction among the photographers A, B, and C. Moreover, we see that work produced by a given artist is itself clustered around a relatively small volume of the full aesthetic space. "A" is obviously confined to one region, separate from (in this case) the volume of space occupied by "B," and both are distinct from the volume occupied by "C."

My point here is not that a feature space within which such a decomposition is possible exists - it may, or may not, for a given set of artists; but only that it suggests an interesting and deep question about what such a set of features - that simulataneously minimizes the differences among a given photographer's works and maximizes the distinction among the works of different photographers - might actually look like! I suspect it may not be like anything we would intuitively expect; if our intuition is anything like what we learn in the standard art and graphics design books. I doubt very much whether the "core features" would include such standard-issue measures as "contrast" and "tone" (though they may very well these). I wonder, too, at just how far separated the artist's "oeuvre clusters" can be made to be, while the spread of each artist's own cluster of works is simultaneously minimized.

One can play other thought games too, of course, For example, having defined some aesthetic space, and having plotted a given artist's current oeuvre - say, what the artist has produced during the last five years of work - we can trace how the artist evolves, using the first plot as a reference. Does the work remain more or less in the same "cloud" of points, so that the artist does not stray too far from his (possible innate?) aesthetic? Or does the cloud slowly dissipate, and reform in another region of the same aesthetic space? Or does the cloud diffuse outward to fill most, or all, of the "old" aesthetic space, thus suggesting that a new feature space - some F'' - exists, and in which the same artist's evolving oeuvre again assumes a cloud-like form?

Thought Experiment #2. Here is an even deeper question; and, truth be told, the real object of my rambling quest. Suppose we have managed to find a special "core aesthetic" space that does precisely what our thought experiment imagines. That is, imagine we have an aesthetic space defined by a special set features (whose relevance, for the moment, is confined solely to photography) that both maximizes the difference between different photographers, and - simultaneously - minimizes the differences between individual photographs of a given photographer. Suppose, further, that we carve out of that space a special set of photographs (and, by association, a special set of photographers) which maximize - for lack of an objectively better-defined word - photographic beauty. Now, imagine we do exactly the same thing (i.e., play the thought experiment as described above) for all of the different kinds of creative endeavors that exist: music, sculpture, literature, mathematics, physics, ... The analog of (generic) "beauty" in art or photography might be - in the case of mathematics, for example - "truth" (as in the truth of theorems); in physics, "beauty" may be aligned with "physical laws" (the "truths" of nature), and so on. What is the underlying meta-pattern that connects the patterns?

Here is my question (and I'll stop at this point): might there be a "universal aesthetic meta-map" that transforms the set of features of one aesthetic space (that describes art, say) to another set of features that describe a different aesthetic space (mathematics, say) but which leaves the measure of "beauty" that is appropriate for each kind of space invariant?
"We do not want merely to see beauty...we want something else which can hardly be put into words; to be united with the beauty we see, to pass into it, to receive it into ourselves, to bathe in it, to become part of it." - C. S. Lewis
"Beautiful" art or music, "physical laws" in physics, and "theorems" in math may be - in a truly fundamental sense - indistinguishable, but only if the analog of "beauty" is correctly defined , and interpreted, in each respective space. Indeed, I suspect that if only we were clever enough creatures to be able to simultaneously apprehend and reflect upon vast multidimensional features spaces, it would only be a matter of "shifting our perceptual / aesthetic axes" (so to speak) for us to be able to transform our endeavors from one creative space into another. Imagine being able to "prove a mathematical theorem" by working on the problem as though it were an art project (and the object of which - in the art space - is to produce a "beautiful work of art"). But whatever space we happen to find ourselves in at a given moment, the object of our quest (and the ultimate arbiter of our creative progress) remains indefagitably the same: truth.

Postscript #1. The way I presented my thought experiment, a (God like) external agent is needed to view the universe of artists and their work to construct (and plot the creative progress in) a D-dimensional aesthetic space. In fact, one can argue that each artist (indeed, each living being) is doing precisely that, ceaselessly, tirelessly, throughout its existence. We are all seeking to be as distinct as possible from all other living beings, even as - at the same time - we desire to be be as integrated into our local cultural / creative fabric as well. It is this insoluble yin-yang tension that drives all self-motivated dynamics; and perhaps all creativity. This fundamental idea of the universe consisting of simultaneous and seemingly antithetical tendencies of integration and distinction (or assertiveness), at all levels of a multidimensional hierarchy, was introduced by author / philosopher Arthur Koestler in a book called Janus. He called all such creatures that strive to do this holons.

Postscript #2. The idea that there is a core universality that underlies all forms of art - all life - is certainly not born in this humble blog entry. In fact, much of my thinking on the subject derives from, and has been shaped by, a magnificent four volume work called Nature of Order by architect / visionary Christopher Alexander (about whom I've written before on my blog).

Postscript #3. A similar idea to the one presented above as thought experiment #1 (but in the context of cosmology) - and developed more completely on a semi-rigorous mathematical level - was proposed a few years ago by physicists Julian Barbour and Lee Smolin. They called it extremal variety. Barbour has published another article on this subject in the Harvard Review of Philosophy.


Anonymous said...

Your concept is intriguing; I feel fortunate to have the background to appreciate it. But I do have a major objection: you seem to assume that there exists a more or less objective "feature space" that might function as you describe. Yet I believe that the evaluation of artworks--not only their scoring on various feature dimensions, but the nature of the dimensions themselves--is a somewhat subjective thing. If so, your musings might apply to each viewer's aesthetic space on a one-by-one basis, but be difficult to reconcile overall. On the other hand, perhaps we share some cultural proclivities.

To me the somewhat astonishing fact is that, as you point out, humans are fairly capable, with study, of recognizing and distinguishing among many artists. Though maybe it's not so surprising as it seems at first blush: with only a dozen features, and allowing a score of high, medium, or low on each, that's potentially enough to separate half a million (3^12=531,441) artists from each other, if each pair differs along at least one of those 12 dimensions. (I'm ignoring the variability within a single artist's production.)

As a case in point, I just noticed that my very recent "Dark Water" series has a significant amount in common with some of your "Process" portfolio. But, I suppose fortunately, I don't think anyone would have too much trouble distinguishing after s little bit of observation. The provocative thing here is that we have similar academic backgrounds. Artistic determinism?

P.S. Regarding contrast and similar differentiators: you might be interested in papers related to this work on photographic style.

Anonymous said...

See Rudolf Arnheim's work

Andy Ilachinski said...


Thanks for your thoughts and the link to what looks like a fascinating paper. As you say (and I don't dispute), the feature space I describe is more fanciful than real - there is simply too much "subjectivity" to be captured by objective manner. My interest, though, is more to use this "concept" as a conceptualization of what such a space *might* look like. And, as you point out, we *do* have a fuzzy impression of it via some quasi-objective features we've been able to identify and articulate.

Note that Christopher Alexander (whom I mention in a postscript), in his magnum opus (which is truly cosmic in scope!) *does* suggest - extremely provocatively - that heretofore purely subjective measures of "beauty" have an objective dimension. That is a radical suggestion to be sure; but science (and art) often owes progress to just such seemingly "absurd" proclamations.

Anonymous said...

Hi Andy,
This is a most thought-provoking post. I confess to having read it multiple times not so much to decipher its meaning but rather to suss out the intent. Unlike Steve Durbin above I do not possess the background (artistically, academically or scientifically) to fully understand the details of your "problem". However the question you raise about a "universal aesthetic meta-map" reminds me of the quest sought by alchemists of old. The transmutation of one thing into another, in your case, the transmutation (albeit in mental terms) of beauty into truth. I'm not sure if I am on the right path here but that is the thought that came to mind.
To me, the question seems like a koan and perhaps that is intended, after all you start off by comparing the problem to "a transient stepping-stone". You state that the answer lies in finding "an objective language (or, least as objective a language as possible) to describe the subjective propensities of, and differences between, individual painters, musicians, or photographers." But is this not quite unlikely, the "objective language" part I mean? Language by nature, is dualistic or subjective. Even if you could come up with some words or parameters approaching objectivity they would only at best be objective to one or to some but never to all. In other words I would say that the answer cannot, in reality, be found either in language or in the mind. So this puts the question back into koan territory.And with that in "mind", Christopher Alexander whom you say suggested that "purely subjective measures of 'beauty' have an objective dimension" gives a perfect paradox to the whole problem. Because a koan is often paradoxical in nature in order to help us alter our perception of reality. As a paradox your problem makes no sense and as a koan it tells us that it is how we think that makes no sense. My guess is that the mind must see the problem more simply before it will cease to be a paradox.

Oh... and "timeless path toward gradual self-enlightenment"... that's funny :)

Andy Ilachinski said...

Cedric, as is becoming your meta-pattern ;-) your insightful comment both goes to the heart of the blog post and uncovers a (not-quite-invisible) few meta-Koans scattered throughout.

Of course, the difficulty with all of this (the post, life, and the univserse), is the impossibility of truly objective discourse. Our entire lives are but pointers (and sometimes, equally as paradoxically, stronger pointers for *others* than ourselves), as we wrestle with what *we* mean, struggling to find words where none can possibly exist.

Still, I cannot help but wonder, whether it is my deliberately willful *act* to fathom an aesthetic meta-pattern that is itself what renders me incapable of apprehending "it" (the "truth", the "meta pattern"), or whether there is truly an uncrossable chasm between (what we ordinarily think of as) subjective and objective worlds. In a way that is very reminiscent of quantum uncertainty among complementary variables, a "scientist" may indeed be capable of penerating deep into an "objectively aesthetic space," but - precisely in so doing! - will also be blind to the otherwise "obvious" subjective truths even casually visible to the shaman.

Anonymous said...

What is the underlying meta-pattern that connects the patterns?


tkh said...

I followed a link here from The Online Photographer, and I'm very impressed. Very thoughtful posts.

Your thought experiments reminded me of Paul Churchland's The Engine of Reason, The Seat of the Soul, where he explains a neural net program, I believe, that was created to learn how to recognize faces from photographs and do some sort of sorting procedure on them. The axes along which the computer learned to sort faces were not at all what a programmer might have chosen, and were difficult to interpret. You could see the images that the computer used as endpoints, but they wouldn't have been very helpful for any human needing guidance on the sorting task. I think Netflix is running into the same sort of issues with their recommendation engine where the categories we might intuitively sort with are not what ends up working. It could be that the feature set that defines beauty in photography is not one that is usable (if it exists). So we could feed pictures into our neural-net beauty identifier and have it confirm for us that something is beautiful, but we still wouldn't really have a recipe for beauty.

It also seems like artistic beauty and mathematical truth may be importantly different in that you can have degrees of beauty but not of truth. You can have degrees of mathematical elegance, I guess, but your proof either works or it doesn't, right? If true, that might complicate any translation.

The subject is very interesting, but it does also strike me as pretty implausible that beauty would be objective enough to do this sort of work. I think of the chapter titled "A Short Dictionary of Misunderstood Words" in Kundera's The Unbearable Lightness of Being, in which he explains what two of his characters mean by "woman," "cemetery," etc, and how those meanings are so shaped by their very different backgrounds such that they end up having very different interpretations of their shared experiences and seriously misunderstand each other.

Finally, have you seen this article (or one like it) about Horace Brock, who claims that beauty, at least in some areas, can be mathematically defined?

Anonymous said...

I may be over simplifying your problem, hocwever I understand it as "What set of features is capable of both minimizing the differnce in individual works on one artist, while maximizing the differences between the work of more than one artist?" If that correctly states the problem, the answer is, the artist(s). The apature in which the artist is defined varies depending upon the discipline.; shorter for photographers than for painters or composers.Furthermore, view humans as a quantum system decohereing into classical manifistations, (Kauffman, "Reinventing the Sacred.") each work is no more than an articulated probability among a wave function of probability. Consequently, the result of the solution to your problem sits along-side a myriad of adjacent possibilities, any one of which may be the primary resourse of the next work, making one wonder how meaningful the exercise.

Howard Goldson,

Anonymous said...

Your whole dream gets blown to smithereens, by the fractal-nature of Universe:

The laws of physics that are "visible" at the nuclear scale, aren't electro/chemistry: they are nuclear.

The laws of physics that are "visible" at the galactic scale are neither nuclear, nor electro/chemical: but are different, again, and less "personal".


The concept of "beauty" in Joe McNally's mind is oft horrifying, to me, and what I know to be beauty would likely be alien / incognizable to him.

What he considers *funny*, however...

( photoproject )

or this:

I love this one...

Found it!

sometimes, he gets it sooo right.

To me...

Therefore, any "dimension" that differentiated some artist from others, would be useless in other comparisons.

Any "dimension" that differentiated "beauty" from "opposite-of-beauty" would be valid in some comparisons, only, and invalid in others.

I consider tranquil vitality to be beautiful.

I consider abstract elegance to be beautiful.

I consider living/spiritual Totality to be beautiful.

Some consider abuse to be beautiful.

Others consider sickness & death to be beautiful.

Not only that, but we're looking at a teensy subset of the animals on this one world.

What's beautiful to a crustacean?

What's beautiful to rabies, or rabies-deformed-brain/mind?

That's the problem with trying to find Universal Subjectivity, it's either Universal, or it's Subjective!


PS, if you want a method that's 5x as fast as Rinzai sect practice, for gaining zen, then use The New Drawing on the Right Side of the Brain to get into R-Mind, and meditate/train IN THAT mode: Totality is made of it, and *real* meditation can be, in that mode. Progress, from there is phenomenal.

It *isn't* a "Mere Coincidence(tm)"
( scientism's Magic Dismissal -- if mind arranges something, is that Mere Coincidence? if one's religion insists that Consciousness isn't real, instead is "The Hard Problem", then... )
that meditation evolved in kanji & visual-symbol-language cultures, but had to be *brought in* to the left-brain-dominant ones, because it couldn't arise in the crippled "knowing" that words-mind is.

Experience it: you'll understand what I'm getting at.



Anonymous said...

Isn't asking "is there a universal aesthetic meta-map" like begging "tell me about tao"? The more deeply one asks about universal realities, the more likely it is that any answers won't be universally true. "The tao that can be told is not the eternal Tao", and all that. (Stephen Mitchell's version)

Aligning the above thought with the other commentators' thoughts about subjectivity and ilachina's mention of quantum uncertainty, I have a prospective definition to float:

subjectivity - that which emerges in others when a universal invariant manifests in oneself

trace said...

Andy, If you haven't heard it, you'd love the audio series "Joseph Campbell on James Joyce - Wings of Art"
It is a fascinating talk on Joyce's theory of the aesthetic. I think the quotes Campbell has on Art in general are some of the best I've heard/read.