Mathematics and the BrainThis is Mathematics Awareness Month, and this year's theme is Mathematics and the Brain...
The volume of a typical human brain is 1400 cm3 and its weight is about 1000 times that of a typical rat's brain. Mundane facts about brain data do little to give us insight into this amazing organ. What is there about the human brain that seems to make possible so many wondrous things that humans can accomplish: oral and written language, producing of art and music, and doing mathematics. Or is it really possible that dolphins have already resolved the status of the Riemann Hypothesis but we are unaware of it because we have found no way yet to communicate with them?
One of the reasons the brain was a mystery for so long is that it is encased in the skull. Under ordinary circumstances one does not have easy access to seeing the brain. Being able to see into the human body took a big leap forward when Wilhelm Roentgen discovered x-rays. Shortly after his discovery of x-rays physicians were already using x-rays to see our bones. However, being encased by the skull, the brain, and other soft tissues were still unknown territory for physicians and physiologists. With contributions from science, engineering and mathematics, physicians have a growing array of imaging tools for seeing inside the skull. At first these brain imaging techniques involved 2-dimensional cross sections of the brain. Many of these techniques are identified with the field of tomography. The Greek word for a "cut" or "section" is "tomas." (Tomography has applications in fields other than medicine., for example, geology.) As the field of tomography has matured, it has been possible to generate three-dimensional medical imaging systems for the brain. These systems allow static views of the brain; increasingly it is also possible to make movies that show the dynamic behavior of the brain. The systems that are now available both to medical practitioners and researchers are:
(A CT image of the brain. Image courtesy of Dr. Keith Johnson, Harvard Medical School.)
(An MRI image of the brain. Image courtesy of Dr. Keith Johnson, Harvard Medical School.)
(A SPECT/PET image of the brain. Image courtesy of Dr. Keith Johnson, Harvard Medical School.)
Images of the kind shown above have become part of the lexicon of our society. They show the brain using three different techniques that physicians have come to rely upon for insight and diagnosis.
The human visual system
One burgeoning area connecting mathematics and the brain is visual phenomena. Human visual information is processed by the brain and uses our amazing eyes. Mathematicians have been involved in a wide variety of problems involving vision:
Illusions and impossible figures
One interesting connection between mathematics and the brain is the reaction many people have to symmetric patterns and patterns which are deceptive (visual illusions and impossible figures). Many individuals including mathematicians find symmetrical designs very appealing. For example, the works of M.C. Escher (1898-1972) hit a very sympathetic chord in the mathematics community.
(Courtesy of Eric Weisstein. Weisstein, Eric W. "Penrose Triangle." From MathWorld--A Wolfram Web Resource.
Other mathematical windows on the brain
The small sample of ways that mathematics interacts with our knowledge of the human brain barely hints at the myriad of interfaces of these two subjects. What follows is just an additional sampler of ways that mathematics is interacting with and providing a valuable tool for attempts to understand the brain.
To the best of our knowledge humans are the only species that does mathematics. Attempts to verify that other animals (plants?) can "do" mathematics have so far not encouraged the belief that any mathematics is being produced by other species. The famous story of "clever Hans," a horse whose master had convinced himself that Hans could do mathematics turned out to be not a case of a horse with mathematical talent but a human master who was sending signals to the horse without realizing he was doing this. What is unique about the human brain that makes this doing mathematics possible? Brain imaging techniques are being used to chart the differences in how humans might do calculations, visualize a polyhderon, try to prove a theorem, or brain storm a problem in number theory. What are the ways that having a "good memory" makes mathematics easier? Many mathematicians describe themselves as having a good memory. What can be said is that mathematicians are using their brains and mathematics to further mathematics' domains and to get further insight into how mathematics gives insight into brains!
Anderson, S., and D. Burr, Spatial and temporal selectivity of the human motion detection system, Vision Research 25 (1985) 1147-1154.
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These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
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