Studies and insights into the nature of consciousness always get my attention. Inevitably I see mathematics in the discussion, tangentially or directly (as with Giulio Tononi’s qualia space). I’d like to outline, here, a particular train of thought that emerged after reading a couple of articles and a few papers.
The first of these, written by psychologist Nicholas Humphrey, appears in a current issue of Scientific American Mind. Consciousness as Art is the title of the article. Humphrey took note of the debate among theoretical psychologists, where ideas seem to fall within one of two perspectives:
Some assert that the manifestly eerie and ineffable qualities of subjective experience can only mean that these nonphysical qualities are inherent in the fabric of the universe. Others, including me, are more suspicious. They argue that consciousness may be more like a conjuring show, whereby the physical brain is tricking people into believing in qualities that don’t really exist.
I’m not sure how any structure brought about by the relationship of our bodies with everything else can be said to not exist. While it may be difficult to find color outside of our own interaction with light, it would seem that deleting its existence wouldn’t help us understand things any better. The perspective of the illusionist is grounded in what many believe is an irreconcilable gap between the physical world and the worlds created by consciousness – the worlds of individual experience and ideas. I’m more interested in finding clues to how these worlds are united (and, I suspect that mathematics is one of our best clues). I suppose I belong to what Humphrey calls the realist camp:
In their view, if your sensations appear to have qualities that lie beyond the scope of physics, then they really do have such qualities. And these realists explain their reasoning by suggesting that the brain activity underlying sensations already has consciousness latent in it as an additional property of matter—a property as yet unrecognized by physics but one that you, the conscious subject, are somehow able to tap into.
I wouldn’t put it that way but, more interesting is what Humphrey proposed as an interesting way to get around the idea to which he subscribes – that the brain is tricking us.
…might it be more persuasive if we were to talk about qualia as art rather than illusion? I am not proposing an alternative theory to illusionism, but my hope is that shifting the emphasis in a positive direction may in fact make the illusionist theory more scientifically acute and at the same time more humanly agreeable.
Thus, this way of thinking about sensations allows us to look out for—and celebrate—the psychological growth that human beings derive from participating in the self-made show.
The chief scientific bonus of conceptualizing consciousness as art may prove to be precisely this: that it raises new questions for an evolutionist about the value and purpose of consciousness. If sensations are art, the artist behind them is actually not the individual brain as such. Rather the artist—the ultimate designer—must be the evolutionary forces of natural selection, which have contrived to put in place the genetic code for building the qualia-generating brain.
This is, I believe, a move in the right direction. Although Humphrey’s proposal for the evolutionary purpose of this is unnecessarily pragmatic. He considers that the evolutionary function of brain art is to “induce you to fall in love with yourself and to encourage you to think of “all humans as equally touched by magic,” to support, I gather, our survival.
There was a reference in this article to a 2008 piece by Christof Koch in which he discussed the work of Martin Giurfa of the University of Toulouse in France who, along with colleagues, published a paper in Nature with the title The concepts of ‘sameness’ and ‘difference’ in an insect. Their abstract tells us this:
..research has indicated that bees are capable of cognitive performances that were thought to occur only in some vertebrate species. For example, honeybees can interpolate visual information, exhibit associative recall, categorize visual information, and learn contextual information. Here we show that honeybees can form ‘sameness’ and ‘difference’ concepts. They learn to solve ‘delayed matching-to-sample’ tasks, in which they are required to respond to a matching stimulus, and ‘delayed non-matching-to-sample’ tasks, in which they are required to respond to a different stimulus; they can also transfer the learned rules to new stimuli of the same or a different sensory modality. Thus, not only can bees learn specific objects and their physical parameters, but they can also master abstract inter-relationships, such as sameness and difference.
Koch highlights some of the specifics in his article:
Although bees can’t be expected to push levers, they can be trained to take either the left or the right exit inside a cylinder modified for the DMTS test. A color disk serves as a cue at the entrance of the maze, so that the bee sees it before entering. Once within the maze, the bee has to choose the arm displaying the color that matches (DMTS) or differs from (DNMTS) the color at the entrance. Bees perform both tasks well. They even generalize to a situation they have never previously encountered. That is, once they’ve been trained with colors, they “get it” and can now follow a trail of vertical stripes if a disk with vertical gratings is left at the entrance of the maze. These experiments tell us that bees have learned an abstract relation (sameness in DMTS, difference in DNMTS) irrespective of the physical nature of the stimuli. The generalization to novel stimuli can even occur from odors to colors.
Koch remarks that, although these experiments do not demonstrate that the bees are conscious, they do caution us to not too quickly reject this possibility.
Bees are highly adaptive and sophisticated creatures with a bit fewer than one million neurons, which are interconnected in ways that are beyond our current understanding, jammed into less than one cubic millimeter of brain tissue. The neural density in the bee’s brain is about 10 times higher than that in a mammalian cerebral cortex, which most of us take to be the pinnacle of evolution on this planet.
In a paper that Koch coauthored with Giulio Tononi they suggest an approach to the study of consciousness that is very promising.
Indeed, as long as one starts from the brain and asks how it could possibly give rise to experience—in effect trying to ‘distill’ mind out of matter, the problem may be not only hard, but almost impossible to solve. But things may be less hard if one takes the opposite approach: start from consciousness itself, by identifying its essential properties, and then ask what kinds of physical mechanisms could possibly account for them.
The paper describes Tononi’s integrated information theory of consciousness in great detail. The essential properties of consciousness that are proposed have a mathematical character:
Taking consciousness as primary, IIT first identifies axioms of experience, then derives a set of corresponding postulates about its physical substrate. The axioms of IIT are assumptions about our own experience that are the starting point for the theory. Ideally, axioms are essential (apply to all experiences), complete (include all the essential properties shared by every experience), consistent (lack contradictions) and independent (not derivable from each other).
Giurfa’s observations of honey bees identify cognitive abilities that are also associated with mathematics – as bees are observed to “master abstract inter-relationships, such as sameness and difference.” My point here is simply that mathematics may provide significant support to the investigation of cognition and/or consciousness in living things. Rudimentary mathematical forms serve as maps to cognitive structure and conscious experience in lives other than our own. And mathematics, as an efficacious cognitive event in our experience, can perhaps alter the terms of the debate about human consciousness between the realists and the illusionists. Mathematics is uniquely important to both physical law and the pure creativity of exploring precisely defined abstract relationships.