Sunday, April 22, 2007

Toward an Aesthetic Grammar: Part I

A few years ago, I gave an invited presentation at the Smithsonian Institute in Washington, DC, entitled Nature's Way: The Art of Seeing Complexity. My lecture was part of a multiweek workshop sponsored jointly by the Washington Center for Complexity & Public Policy and the Resident Associate Program of the Smithsonian Institution.

The ambitious goal I set out to accomplish with my talk - which, in hindsight I ought to have known would be impossible to achieve in the short time I had to achieve it (about two hours) - was to use the soul-searching inner musings of a physicist as photographer as a springboard toward forging a possible conceptual bridge between art and science; one that is defined by an aesthetic grammar, and hints at an even deeper aesthetic physics (two phrases that I promise to define more carefully below). As I diligently plowed through my slides, and talked through a few I had prepared especially to explain these subtle points, I could tell from the many blank stares and questioning smirks, that my skeletal new art-science "aesthetics theory" was destined to fall far short of my intended goal that day.
"The division of the perceived universe into parts and wholes is convenient and may be necessary, but no necessity determines how it shall be done." - Gregory Bateson (anthropologist, 1904 - 1980)
So, for another, slightly expanded attempt at communicating some soul-searching inner musings of a physicist as photographer...let me begin - in Part I of a multipart series of essays I intend posting in the coming weeks on the same topic as my Smithsonian talk, but retitled Towards an Aesthetic Grammar - by introducing a provocative theorem that I will first make a cautionary meta-claim about: please be forewarned that the theorem I am about to state will likely strike you either as obvious (at best) or idiotically vacuous (at worst). However, I will immediately argue that not only does the truth (of its interpretation) lie nowhere near these two extremes, but that the theorem is deceptively subtle and points to a universal "core truth" that underlies all cognitive, scientific and creative endeavors!

What is this remarkable theorem? It is called the "Ugly-Duck Theorem" (named after the well-known story by Hans Christian Andersen), and was proposed and proven by statistician Satosi Watanabe in 1969 (who was then at the University of Hawaii).

Suppose that the number of predicates that are simultaneously satisfied by two nonidentical objects of a system, A and B, is a fixed constant, P. The Ugly Duck theorem asserts that the number of predicates that are simultaneously satisfied by neither A nor B and the number of predicates that are satisfied by A but not by B are both also equal to P. While this assertion is easy to prove, and certainly appears innocuous at first glance - indeed, you would be forgiven to think it entirely "meaningless" since it is merely restating an obvious combinatorial fact about the set of possible predicates - it has rather significant philosophical and conceptual consequences.
"Thought is creating divisions out of itself and then saying that they are there naturally." - David Bohm
For example, suppose that there are only three objects in the world, arbitrarily labeled (@,@,#). An obvious interpretation is that this describes two kinds of objects: two @s and one #. But there are other ways of partitioning this set. For example, line them up explicitly this way: @ @ #. An implicit new organizing property seems to emerge: the leftmost @ and the rightmost # share the property that they are "not in the middle". We are free to label this property using the symbol @, and the property of being in the middle, #. Now, substituting the new property for each of the original objects, we have @ @ # -> @ # @.

Had we sorted these three objects according to the new property (that discriminates according to spatial position), we would again have two kinds of objects, but in this case they would have been different ones. Obviously, we can play this game repeatedly, since there are endless number of possible properties that can arbitrarily be called @ and #. That is the point. Unless there is an objective measure by which one set of properties can be distinguished from any of the others, there is no objective way to assert that any subset of objects is better than, or different from, any other.

The theorem demonstrates that there is no a priori objective way to ascribe a measure of similarity (or dissimilarity) between any two randomly chosen subsets of a given set. (Or, stated more whimsically, the theorem states that, all things being equal, an ugly duck is just as similar to a swan as two swans are to each other!) More technically speaking, we see that asymmetries within a system (i.e., differences) can be induced only either via some externally imposed “aesthetic” measure, or generated from within.


"Of course" ... you might be saying ... "that is obvious! But why is this important?" It is important because it demonstrates that - fundamentally - all of our perceptions of the world, precisely because they are demonstrably not all uniform, appear as sets of different things interrelated in a myriad of ways because of an internal aesthetic (or internal grammar, or physics!) that we automatically impose on what we perceive (doing so mostly unconsciously). The problem is to find a way to characterize and articulate what such a grammar might actually look like!


We "see" rocks and chairs and people primarily because nature has evolved an immeasurably powerful sensory-cognitive processing mechanism that rapidly "tags" for us (for our "I") the patterns in our environment that we will most likely be interacting with repeatedly throughout our lifetime. These objects are not visible to us (as "things") because the universe has labeled them "objectively meaningful" in a global sense (I doubt whether the universe really cares whether a particular transient pattern of atoms is called a "chair", a "collection of wooden planks" or "an exemplar of post-modern, neo-minimalist drivel"); rather, they appear to us as "meaningful" only because they are meaningful to us locally, in terms of the natural aesthetics we were born with (and evolve for ourselves as we interact with our perceptions and experiences) that determine what objects we can see, and the degree to which we can distinguish one object from another.

Who we are - our "I" - is defined and shaped most strongly by our internal aesthetic; which, I shall argue shortly, does not just describe "what we happen to think is beautiful at the moment" but molds our entire conception of the world, with all of the artistic, scientific, philosophical and spiritual depths that entails.

When I use the phrase "conceptual grammar" (or "aesthetic grammar") I mean - no more and no less - the set of aesthetic-weights we use (mostly unconsciously) to ascribe more or less "thingness" to an object A compared to another object B. According to the Ugly Duck theorem, we would expect the components of this set of weights to all be equal and therefore completely undiscerning in a rigorously objective world. Our conceptual grammar, understood in this way, therefore also constitutes the backbone of a primitive "local physics" we all use to describe our world; where by "physics" I mean a set of "organizing principles" that describe the underlying patterns of what our aesthetics "permit" us to recognize as existing.

Thus, when I write "grammar", I am thinking of primitive building blocks of "things" that (we imagine and/or perceive to) populate our (aesthetically generated asymmetric) local world, and the ways in which things may be "combined" to yield other things. And when I write "physics", I am thinking of the primitive building blocks of "patterns" that connect the things.
"Man tries to make for himself in the fashion that suits him best a simplified and intelligent picture of the world; he then tries to some extent to substitute this cosmos of his for the world of experiences, and thus to overcome it. This is what the painter, the poet, the speculative philosopher, and the natural scientist do, each in his own fashion." - Albert Einstein
I will discuss some important consequences of the Ugly Duck theorem, and suggest how it might be used to generalize what we (think we) know about our "scientific aesthetics" to begin probing what an (objectively artful) "aesthetic grammar" may look like, in Part II (stay tuned....) Speculations on what all of this has to do with complexity, photography, the "art of seeing", and using art to find one's "I", will also appear in forthcoming essays.

Technical Note: The Ugly Duck Theorem complements another well-known theorem called the No Free Lunch theorem, proven by Wolpert and Macready in 1996. The No Free Lunch theorem asserts that the performance of all search algorithms, when averaged over all possible cost functions (i.e., problems), is exactly the same. In other words, no search algorithm is better, or worse, on average than blind guessing. Algorithms must be tailored to specific problems, which therefore effectively serve as the external aesthetic by which certain algorithms are identified as being better than others. Technical proofs of Watanabe's theorem appear in his books Knowing and Guessing and Pattern Recognition (both of which are, sadly, out of print).

3 comments:

chuck kimmerle said...

Of course" ... you might be saying ... "that is obvious!

I can honestly say that the term "obvious" did NOT come to mind :)

I think I have to reread it a couple more times for a more complete understanding, but find it quite interesting and thought provoking.

David S. Mazel said...

Well, well, well. My sense is that any grammar you might propose is just as good as any other grammar. How you, a physicist, may see a "pattern" or grammar, is not how anyone else might see it. That is, the ugly ducking theorem applies not just to objects, but to everything, even to itself.

I suspect that what you have here, as you state albeit with twists, turns, and verbiage, is that all things are subject to interpretation by the observer and the observer is free to impose his values upon what he observes.

Thus, a wooden chair is so classified by creatures with butts, but other creatures may see them (the chairs, not the butts) as firewood. No other term need apply, no classification necessary.

I am reminded, often, by what a past graduate school professor (Dr. Russell Mersereau) told me: When people say their algorithm is optimum, what they mean is that they have defined a certain function and their algorithm optimizes that function. Thus, each engineer's algorithm is optimum, but each applies to different functions.

The Ugly Duckling Theorem, by the way, should be taught to all teachers in elementary school and higher. How often, and I am on a slightly different subject, is a parent confronted with homework where his child is asked to find the next entry in a given series? Say the series is: 1, 2, 3, ____? Is 4 the correct answer? Maybe. But maybe 5 works just as well if we think of prime numbers and allow 1 to be prime. Or, maybe 8 is correct, if your street address number is 1238? Who can say, without an external metric, which number really belongs. It's impossible and yet we teach children that it is not only possible, but there is a single correct answer.

Frankly, I don't know how far you, Andy, will get with a grammar, but I do think that such a discussion is long overdue. Not just for artists and physicists, but for all of us who think we see a pattern that is really self-induced and without any more merit than any other pattern.

Steve Durbin said...

David,

I have just a slightly different take on where this is going, or where it might go if I were writing it. I would de-emphasize both the observer's freedom of interpretation that you mention and Andy's "natural aesthetics we were born with" in favor of how our minds develop "as we interact with our perceptions and experiences." I don't think we evolved to recognize chairs -- though some things, such as faces, are doubtless special. In my view, your personal segmentation optimizing function essentially is, for purposes of this grammar, your life summed up statistically. For example, if you're a botanist, you're more likely to see that clump of bushes as a set of distinct species. To the extent we share experiences (culture), there are things we'll all recognize and care about. Some patterns may not have more intrinsic merit, but they may have more significance if they relate to the experience of more people. As for getting eventually to "value," going from optimizing segmentation or information to optimizing pleasure, that would seem to be a trickier move. An interesting one to think about, I agree. But will it help me make better photographs?