An article in the December issue of Scientific American gave me a new insect behavior to ponder and one that might reveal, in the insect’s biology, a distant cousin to the mathematical idea we call mapping. It seems that there are insects that have a talent for recognizing faces. Their talent has much in common with our own facial recognition abilities, except for the brain that carries it. It was a young graduate student, researching the social lives of a particular kind of wasp, who first noticed that she could distinguish individual wasps by their facial markings. Elizabeth Tibbetts (now an Associate Professor at the University of Michigan) then wondered if the wasps identified each other by these markings and began research directed at answering that question. What she found was that the wasps did respond to changes in facial markings and did use variation in facial patterns for individual recognition. Tibbetts further investigated this aspect of their behavior by trying to train them to learn to differentiate among patterns other than their own facial markings. The wasps quickly learned to accurately select among wasp faces, but had noticeably more difficulty discriminating among the other images. This is taken as strong evidence that wasps have neural systems specialized for wasp face recognition. Although honeybees do not use face recognition mechanisms in their own daily lives, researchers have successfully trained them to discriminate human faces. And further,
…although the bees learned faces slowly as compared with P. fuscatus wasps and humans, they were able to develop some ability to process faces holistically, even though they are not hardwired to do so, as P. fuscatus wasps and humans are. Second, honeybees were able to learn multiple view points of the same face and interpolate based on this information to recognize novel presentations. For example, after a bee learns the front and side view of a particular face, it will be able to correctly choose a picture of the same face rotated 30 degrees, even if it has not previously seen this particular image.
Inherent to facial recognition mechanisms is some idea of spatial arrangement as well as transformation (when presented with multiple views of the same face). So I searched out what has been understood about our own neural systems and found an article that Carl Zimmer wrote for Discover in January 2011. The article describes a model for understanding how our own facial recognition mechanisms work. The idea was first proposed 25 years ago by Tim Valentine and Vicki Bruce, two psychologists at the University of Nottingham, and it has been consistently gaining support. Zimmer reported on the work of cognitive neuroscientist Marlene Behrmann, at Carnegie Mellon University who, with her colleagues, had made some important observations when they set out to compare the brains of individuals who are face-blind to those who are face-sighted.
Valentine and Bruce argued that our brains do not store a photographic image of every face we see. Instead, they carry out a mathematical transformation of each face, encoding it as a point in a multidimensional “face space.” (emphasis added)
On a map of face space, you might imagine the north-south axis being replaced with a small-mouth- to-wide-mouth axis. But instead of three different dimensions, like the space we’re familiar with, face space may have many dimensions, each representing some important feature of the human face. Just as the ancient cosmos was centered on Earth, Valentine and Bruce argued that the facial universe is centered on the perfectly average face. The farther a face is from this average center, the more extreme it becomes.
This is not just a model for describing what happens. It is understood as a way to characterize how the brain can negotiate the infinite possibilities that emerge from our experience.
…it offers an elegant explanation for how we can store so many images of faces in our heads. By reducing a face to a point—creating a compact code for representing an infinite number of faces—our brains need to store only the distance and direction of that point from the center of face space. Face space also sheds light on the fact that we are more likely to correctly identify distinctive faces than typical ones. In the center of face space, there are lots of fairly average faces. Distinctive faces dwell far away from the crowd, in much lonelier neighborhoods.” (emphasis added)
This is exactly how our consciously defined mathematical ideas can operate in our experience. They corral infinite possibilities into well defined concepts whose behavior can be observed and interpreted. In this way it is an arrangement of ideas, that compacts experience and often extends the reach of cognition – of our ability to see and understand our world.
Another kind of learning is discussed in an excerpt from the book Incognito that also appeared in Discover in October of 2011 under the title Your Brain Knows a Lot More Than You Realize. The author of the piece is neuroscientist David Eagleman. Eagleman highlights learning that can only be accomplished with a kind of trial and error training. He describes what has come to be called chick sexing – the process of quickly determining the sex of chicks that have been hatched.
The Japanese invented a method of sexing chicks known as vent sexing, by which experts could rapidly ascertain the sex of one-day-old hatchlings. Beginning in the 1930s, poultry breeders from around the world traveled to the Zen-Nippon Chick Sexing School in Japan to learn the technique.
The mystery was that no one could explain exactly how it was done. It was somehow based on very subtle visual cues, but the professional sexers could not say what those cues were. They would look at the chick’s rear (where the vent is) and simply seem to know the correct bin to throw it in.
And this is how the professionals taught the student sexers. The master would stand over the apprentice and watch. The student would pick up a chick, examine its rear, and toss it into one bin or the other. The master would give feedback: yes or no. After weeks on end of this activity, the student’s brain was trained to a masterful—albeit unconscious—level.
Eagleman also discussed what he called flexible intelligence saying the following:
One of the most impressive features of brains—and especially human brains—is the flexibility to learn almost any kind of task that comes their way… This flexibility of learning accounts for a large part of what we consider human intelligence. While many animals are properly called intelligent, humans distinguish themselves in that they are so flexibly intelligent, fashioning their neural circuits to match the task at hand. It is for this reason that we can colonize every region on the planet, learn the local language we’re born into, and master skills as diverse as playing the violin, high-jumping, and operating space shuttle cockpits.
And a clear reference to mathematics is made in the article’s introduction:
Eagleman’s theory is epitomized by the deathbed confession of the 19th-century mathematician James Clerk Maxwell, who developed fundamental equations unifying electricity and magnetism. Maxwell declared that “something within him” had made the discoveries; he actually had no idea how he’d achieved his great insights. It is easy to take credit after an idea strikes you, but in fact, neurons in your brain secretly perform an enormous amount of work before inspiration hits. The brain, Eagleman argues, runs its show incognito. Or, as Pink Floyd put it, “There’s someone in my head, but it’s not me.
There are a few things going on here. First, I’m intrigued by the presence of what could be called ‘mappings’ in the biology of a creature so different from us. This suggests the possibility that a mathematics-like mechanism is not just the product of mammalian brain processes. I’m also impressed by the similarity between the face space model for face recognition and the conceptual compression that defines mathematical ideas in general. Finally, I’m always interested in the extent to which our sense making and even our mathematical insight happens outside of our awareness.