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This month's topics:
Original math on Futurama
The Matt Groening geek-oriented Futurama series featured some original mathematics recently. Episode 10, "The Prisoner of Benda," (aired on August 19, 2010, watch clips here) features a mind-swapping machine (works like the Tibetan skull in "Vice Versa" except looks more like a dual salon hair dryer) that eventually requires a mathematical theorem to get everyone back in his or her own head. As reported in the super-gadget blog Gizmondo, "In the show, you can only switch bodies once with the same pair of people, so they needed an equation to prove that with enough switching bodies around, everyone will eventually end up as who they really are." Gizmondo tells us that "Ken Keeler, the Futurama writer behind the theorem, actually has a PhD in math" and that his theorem is based on group theory. See the proof, and a "visual breakdown" on the Futurama Fanarama site.
The math of consciousness
"Sizing Up Consciousness By Its Bits" by Carl Zimmer was the lead story in the Science section of the Tuesday, September 21, 2010 New York Times. The article is about Giulio Tononi, a consciousness scientist at Wisconsin, who has a new approach to the field: "He and his colleagues are translating the poetry of our conscious experiences into the precise language of mathematics." The idea is to use a version of Claude Shannon's information theory. "Consciousness, Dr. Tononi says, is nothing more than integrated information. Information theorists measure the amount of information in a computer file or a cellphone call in bits, and Dr. Tononi argues that we could, in theory, measure consciousness in bits as well." We are not told exactly what integrated information is supposed to be, but the measure of it is called "phi", and we have more of it when we are wide awake than when we are asleep.
It is possible to calculate how much phi there is in a network: it turns out to be related to the network topology: "Networks gain the highest phi possible if their parts are organized into separate clusters, which are then joined." Unfortunately, even for as primitive a creature as Caenorhabtidis elegans ("only has 302 neurons in its entire body") the lifetime of the universe, according to Tononi's collaborator Christoph Koch, is not long enough for a precise calculation, at the present state of the theory; but a rough estimate of the phi of C. elegans is in the works.
Zimmer: "If Dr. Tononi is right, he and his colleagues may be able to build a 'consciousness meter' that doctors can use to measure consciousness as easily as they measure blood pressure and body temperature." And in fact they have discovered that after delivering an 0.1-second pulse of magnetism to a spot on the scalp of a volunteer they can measure brain reverberations "like a ringing bell, with neurons firing in a complex pattern across the brain for 295 milliseconds," but that when the subject is given the sedative midozolam, the pulse elicits "a much simpler response in a much smaller region, lasting only 110 milliseconds." The connection with phi is not explained, but the commercial possibilities are clearly sketched out. Finally a light note: the test would not be limited to humans. "Dr Tononi suspects that ... animals will prove to have different levels of consciousness, depending on their integrated information. Even C. elegans might have a little consciousness."
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