In last weeks post, I reported on the work of a computer scientist (Jürgen Schmidhuber’s artificial curiosity) and neuroscientist Gerald Edelman. I would like to follow-up a bit with more about Edelman’s work and perspective, in part because I was captivated by a story he told (in more than one venue) to illustrate the fact that the brain “is not a machine for logic but in fact a construction that does pattern recognition. And it does it by filling in, in ambiguous situations.” I’m quoting from an interview that appeared in Discover. Here’s the story as he told it:
There’s a neurologist at the University of Milan in Italy named Edoardo Bisiach who’s an expert on a neuropsychological disorder known as anosognosia. A patient with anosognosia often has had a stroke in the right side, in the parietal cortex. That patient will have what we call hemineglect. He or she cannot pay attention to the left side of the world and is unaware of that fact. Shaves on one side. Draws half a house, not the whole house, et cetera. Bisiach had one patient who had this. The patient was intelligent. He was verbal. And Bisiach said to him, “Here are two cubes. I’ll put one in your left hand and one in my left hand. You do what I do.” And he went through a motion.
And the patient said, “OK, doc. I did it.”
Bisiach said, “No, you didn’t.”
He said, “Sure I did.”
So Bisiach brought the patient’s left hand into his right visual
field and said, “Whose hand is this?”
And the patient said, “Yours.”
Bisiach said, “I can’t have three hands.”
And the patient very calmly said, “Doc, it stands to reason, if you’ve got three arms, you have to have three hands.”
I found a five-part series of essays by ERROL MORRIS in a 2010 New York Times blog that considers some of the implications of this anomaly. The title of the series was The Anosognosic’s Dilemma: Something’s Wrong but You’ll Never Know What It Is.
These essays use the anosognosic’s experience to highlight the difference between the mysteries we can solve (questions we can formulate but not yet answer) and “unknown unknowns,” questions of which we cannot conceive. For David Dunning, a social psychologist at Cornell, “ignorance profoundly channels the course we take in life. And unknown unknowns constitute a grand swath of everybody’s field of ignorance.” Applied to the sciences, Morris quotes Freeman Dyson (from “What Price Glory? A review of Steven Weinberg’s Lake Views, The World and the Universe,” New York Review of Books, June 10, 2010). Dyson is addressing the possibility of understanding everything when he says:
“I would be disappointed if nature could be so easily tamed. I find the idea of a Final Theory repugnant because it diminishes both the richness of nature and the richness of human destiny. I prefer to live in a universe full of inexhaustible mysteries, and to belong to a species destined for inexhaustible intellectual growth.”
The thoughts that ran through my mind circled around the way that mathematics, as an almost pure study of concepts, has opened up previously inconceivable possibilities – like curved space or the quantum mechanical world.
I listened to Edelman speak in an interview and a brief talk and he has a lot to say about the processes that bring consciousness about, and what seems to be unique about what he calls the “higher order consciousness” of human experience. He begins with the idea that the brain is embodied and the body and the brain are embedded in the world. Perception he describes as the brain carving up the world into something sensible – what we see and what we here. It is with language (in particular the effectiveness of syntax) that we can remove ourselves from the remembered present (primary consciousness) and this creates a higher order consciousness. He often makes the point that the power of this language rests more in its ambiguity more than its clarity because it can describe situations that are not precisely defined. He attributes our sense of individuality to our own sense of our own movements (a proprioception) and acknowledges that we don’t yet understand motor control very well. The anosognosic’s experience he describes as the brain adjusting to what the body is equipped to do. He also made an interesting claim – that evidence of artificially created mental activity (mental activity in an artifact) would be that the artifact could mentally rotate an object, or see the object in the rotation. Rotations always make me think of mathematics.
A few of Edelman’s observations are particularly relevant to how I think about mathematics. One is that family arguments as well as solutions to complex mathematics problems are happening within the same system of combinatorial interactions which he calls the dynamic core of consciousness. These interactions involve signals from the world (both input and output), signals from the body, and the brain speaking to itself through memory, fantasy and imaging. About this he says the combinatorial interactions are “hair-raising” and we don’t have measurement systems that can understand them microscopically. But he seems to regularly associate mathematics with the “precision of computing,” rather than the powerful “ambiguity of language.” I would suggest, however, that the conceptual grounding of modern mathematics shares something with Edelman’s ideas about the power of ambiguity in language, when the structure and range of mathematics’ applicability was enhanced with its very broad generalizations. For Edelman, associativity and metaphors start things off and then computation is applied. But mathematics occurs in both. And I would agree with Dyson. I also prefer “to live in a universe full of inexhaustible mysteries, and to belong to a species destined for inexhaustible intellectual growth.” I often see mathematics as the evidence for, as well as the access to, these inexhaustible mysteries.
Edelman ended one of his talks with this poem from Emily Dickinson and I liked it very much:
The Brain – is wider than the Sky –
For – put them side by side –
The one the other will contain
With ease – and you beside –
The Brain is deeper than the sea –
For – hold them – Blue to Blue –
The one the other will absorb –
As sponges – Buckets – do –
The Brain is just the weight of God –
For – Heft them – Pound for Pound –
And they will differ – if they do –
As Syllable from Sound –
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