Let's look at one more optical illusion which reveals important features of how our brains perform inference, and suggests how better awareness of these processes of inference can lead to improved thinking, including in our daily lives. This time around the theme is ambiguity.
The spinning dancer
The spinning dancer is a remarkable piece of work. Do you see the dancer pivoting clockwise on her left foot? Or counterclockwise on her right? If you're at all like me, you're now seeing one or the other - but if you look at the picture for some time, you'll suddenly see the dancer spinning in the opposite direction. It may help (for reasons I'll explain below) to focus on the pivot foot or thereabouts, mentally blocking out the rest of the image. Can you pick a direction on purpose? (Part of what makes investigating the human mind fascinating hobby is finding out what others' brains can do that mine can't, and vice versa. Initially I had no control at all, but interestingly, over the course of writing this post I got much better.)
Focusing on the foot helps explain how the illusion works: the image provides no clue to distinguish between "toes front" or "heel front", it's all a dark shadow with no depth information. What we see is a foot-shaped cutout growing and shrinking due to a foreshortening effect, alternately to the left and right. This information is perfectly compatible with either direction of spin. (The rest of the image is more of the same; the entire image is functionally equivalent to a dark bar growing and shrinking in alternate directions.) The interesting question is then, why does our consciousness insist on reporting that we're seeing a perfectly unambiguous direction of motion? We're not seeing an ambiguous dancer: we're seeing one that is clearly spinning one way - until the "flip" happens, and we see her just as clearly spinning the other way.
This is reminiscent of the way our explicit models of the world have trouble dealing with quantum uncertainty - our intuition is that things must happen one way or the other, "as if the half-silvered mirror did different things on different occasions", as if the dancer actually was spinning one way then another. In the latter case at least, sober reflection tells us this can only be in the map, not in the territory - the animated GIF isn't being changed right under our noses.
"Perception is inference from incomplete information", says Jaynes - I have noted previously how this may bring insight into where our biases come from. The spinning dancer tells us something about how it feels, from the inside, to perform inference under uncertainty. That is what ambiguity is - a particular kind of uncertainty. Not the one that results from a paucity of information: the dancer illusion works because the image is quite detailed, not in spite of it. Rather, it is uncertainty that results from having too many hypotheses available, and lacking some crucial information to distinguish the correct one among them.
(Abstractly, we see that it is best to be right about something, and worst to be wrong; to be uncertain is somewere in the middle. Our visual system has a different opinion, and prefers the following ranking: right > wrong > uncertain. "Make a decision, even if it is the wrong one, we can always revise it later." Actually, many of our decision processes have that same bias; I'll revisit this topic when I post at greater length about "real options". For now, the topic I'm sticking to is ambiguity as a particular type of uncertainty.)
Last night I shot an elephant in my pajamas. How he got in my pajamas, I’ll never know. - Groucho Marx
It takes some effort to construct ambiguous pictures, whereas anything expressed in words seems to enjoy a head start. This is great news for people with a sense of humor: linguistic ambiguity is a constant source of merriment. In fact, though there are many competing theories of humor, it makes at least some sense to see ambiguity and its related themes of frame-crossing, reinterpretation as playing a key role in humor in general.
To some, ambiguity is much less funny. Some professions, including jurists, proponents of synthetic languages such as Lojban or Loglan, and software engineers see ambiguity as an evil to be uprooted. The most famous cartoon (excepting perhaps some Dilbert favorites posted near every cubicle) among software professionals is an extended lament about the consequences of ambiguity in human language. Ambiguity is the source of unmeasurable amounts of confusion and even hurt among people relying on the written word to communicate - an increasingly common circumstance in the Internet age. Not that oral communication is exempt; but at least in most cases it offers more effective mechanisms for error correction.
Sacrifice, duality, reframing: the powers of ambiguity
And yet, vexing as it may be to acknowledge it, instrumentally effective thinking often seems to rely on ambiguity.
In the ancient game of Go, there is a certain level of play that can only be reached by mastering the art of sacrifice. Now, in some cases this may be part of a pre-established plan: stones that you are defending will get a better position if, as a preliminary, you place a stone within enemy territory solely in preparation for a move that threatens to rescue it. In many instances, though, sacrifice involves redefining a previously valuable stone or set of stones as "sacrifice stones". What is called "light" play often involves deliberately ambiguous moves, in which you have a plan to abandon one of several stones without depending on which one; the opponent's play will determine that.
Another example involves the mathematical theme of duality, where a given kind of structure can be expressed in one of two ways which are equivalent in meaning but rely on different tools or operations. Some problems turn out to be easier to solve in a domain which is the dual of that where they were originally formulated. Unfortunately, my math having gone rusty for quite a while, I don't recall offhand any examples that I feel familiar enough with to discuss here, but to give you a hint of the flavor, consider a frequent "trick" in computing probabilities: when asked the probability of A (say, "at least two of the people in this room share a birthday") it's often easier to consider the probablity of not-A ("no two people share a birthday") and taking the complement.
Another way to exploit ambiguity turns up in the domain of interpersonal relationships, in the guise of "framing" and "reframing". Some of the advice in Alicorn's recent post, for instance, involves casting around for reframes - ways of interpreting behaviour that you dislike in a person that make these behaviours tolerable or understandable instead of irritating and repulsive. In Stumbling on Happiness Daniel Gilbert argues that ambiguity is a key component of psychological resilience. A person's success in life is partially determined by their ability to redefine their values, sense of happiness, etc. on the fly, in answer to the difficulties they encounter. This is only possible if our interpretation of the world contains lots of ambiguity to start with.
As readers of LessWrong, "hobbyists of the mind", we can often gain insight into our human minds by framing questions about constructed minds - by forcing ourselves to confront the design space of possible minds. A corollary attitude is to be cautious about the reverse process - unreflectively projecting some perceived attribute of human minds, such as randomness or reliability, into a necessary property of artificial intelligences. Still, I have come so far in this post in large part to ponder whether ambiguity is a necessary capability of minds-in-general, rather than a human design flaw.
Douglas Hofstadter's Fluid Concepts and Creative Analogies discusses fluidity, a theme that strikes me as closely related to ambiguity (and is explicitly discussed throughout). Hofstadter is fond of "microdomains", simplified settings where human intelligence neverthelesss still easily exceeds what we can, as a rule, program machines to do. A favorite of mine is "Do this!" exercises, which inspired Robert French's Tabletop research program: two people are seated at a typical restaurant or café table, with similar (or very different) implements on each side of the table: plates, forks, knives, glasses... One person touches an item and challenges the other to "do this!".
Ambiguity arises in the Tabletop domain because exact mappings between the two sides may not exist, but "analogical" mappings often do: your wine glass maps to my water glass, for instance. The fun starts when more than one plausible analogical mapping suggests itself. Your lone wine glass maps either to my wine glass (paired with a glass of water) or to my salt shaker (the only non-plate unpaired item on my side).
I suspect that efficient cross-domain generalization requires dealing with ambiguity and analogy. Back in the human mind-space, few theories of the scientific process deal explicitly with ambiguity and analogy, Pickering's mangle being a notable exception. In some vague sense it seems to me that ambiguity provides "degrees of freedom" which are necessary to the conceptions of plans, and their flexible execution - including sacrificing, changing domains or notations, and reframing. I expect it to be a major theme of instrumental rationality, in other words, and therefore would like to delineate a more precise formulation of this vague intuition.
This post has explored some themes I'm likely to return to, or that form some kind of groundwork (for instance when I touch on "programming as a rationalist skill" later on). But more importantly, it is intended to encourage further discussions of these themes from other perspectives than mine.
What do you know about ambiguity?