The Hilbert (axiomatic) method
We place a locked cage onto a given point in the desert. After that we introduce the following logical system:
Axiom 1: The set of lions in the Sahara is not empty.
Axiom 2: If there exists a lion in the Sahara, then there exists a lion in the cage.
Procedure: If P is a theorem, and if the following is holds: “P implies Q”, then Q is a theorem.
Theorem 1: There exists a lion in the cage.
The geometrical inversion method
We place a spherical cage in the desert, enter it and lock it from inside. We then perform an inversion with respect to the cage. Then the lion is inside the cage, and we are outside.
The projective geometry method
Without loss of generality, we can view the desert as a plane surface. We project the surface onto a line and afterwards the line onto an interiour point of the cage. Thereby the lion is mapped onto that same point.
The Bolzano-WeierstraÃŸ method
Divide the desert by a line running from north to south. The lion is then either in the eastern or in the western part. Let’s assume it is in the eastern part. Divide this part by a line running from east to west. The lion is either in the northern or in the southern part. Let’s assume it is in the northern part. We can continue this process arbitrarily and thereby constructing with each step an increasingly narrow fence around the selected area. The diameter of the chosen partitions converges to zero so that the lion is caged into a fence of arbitrarily small diameter.
This type of scientific approach to real world tasks has not completely gone out of style, it seems. The LA Times reports that Thomas W. Gillespie and John A. Agnew, two UCLA professors of geography, et alia, in an article in MIT’s International Review, have undertaken to pin down Osama bin Laden’s current hideout, using biogeographic theory. They may be wrong, but I think we should bomb the buildings they’ve identified just for luck.
While U.S. intelligence officials have spent more than seven years searching fruitlessly for Osama bin Laden, UCLA geographers say they have a good idea of where the terrorist leader was at the end of 2001 â€” and perhaps where he has been in the years since.
In a new study published online today by the MIT International Review, the geographers report that simple facts, publicly available satellite imagery and fundamental principles of geography place the mastermind behind the Sept. 11 attacks against the U.S. in one of three buildings in the northwest Pakistan town of Parachinar, in the Kurram tribal region near the border with Afghanistan
The researchers advocate that the U.S. investigate â€” but not bomb â€” the three buildings. …
The UCLA findings rely on two principles used in geography to predict the distribution of wildlife, primarily for the purposes of designing approaches to conservation. The first, known as distance-decay theory, holds that as one travels farther away from a precise location with a specific composition of species â€” or, in this case, a specific composition of cultural and physical factors â€”the probability of finding spots with that same specific composition decreases exponentially. The second, island biogeographic theory, holds that large and close islands have larger immigration rates and will support more species than smaller, more isolated islands.
Inspired by distance-decay theory, the seven-member team started by drawing concentric circles around Tora Bora on a satellite map of the area at a distance of 10 kilometers â€” or 6.1 miles â€” apart.
“The farther bin Laden moves from his last reported location into the more secular parts of Pakistan or into India, the greater the probability that he will be in an area with a different cultural composition, thereby increasing the probability of his being captured or eliminated,” Gillespie said.
Then, informed by island biogeographic theory, the researchers scoured the rings for “city islands” â€” or distinctly separate settlements of considerable size.
“Island biology theory predicts that he would find his way to the largest but least isolated city of that area,” said Gillespie, an authority on measuring and modeling biodiversity on Earth from space. “If you get stuck on an island, you would want it to be Hawaii rather than one with a single palm tree. It’s a matter of resources.”
The approach netted 26 cities within a 12.4-mile radius of Tora Bora on imagery from Landsat Enhanced Thematic Mapper Plus (ETM+), a global archive of satellite photos managed by NASA and the U.S. Geological Survey. With a 2.7-square-mile footprint, Parachinar turned out to be the largest and fourth-least isolated city, the team determined.
“Based on bin Laden’s last known location in Tora Bora, we estimate that he must have traveled 1.9 miles over a 13,000-foot-high pass into Kurram and then headed for the largest city, which turns out to be Parachinar,” said Agnew, who is the current president of the Association of American Geographers, the field’s leading scholarly organization.
The researchers ruled out cities on the Afghanistan side of the border because the country was occupied at the time by U.S. and international forces and has been particularly unstable ever since.
“The Pakistan side of the border is much better for hiding because of its ambiguous political status within the country and the formal absence of U.S. or NATO troops,” Agnew said.
Faced with the prospect of picking from more than 1,000 structures clearly portrayed in the satellite imagery of Parachinar, the team decided to come up with a short list of the criteria that bin Laden would need for housing, based on well-known information about him, including his height (between 6’4″ and 6’6″, depending on the source), his medical condition (apparently in need of regular dialysis and, therefore, electricity to run the machine) and several basic assumptions, such as a need for security, protection, privacy and overhead cover to shield him from being spotted by planes, helicopters and satellites.
So they looked for buildings that could house someone taller than 6’4″ and were surrounded by walls more than 9 feet tall (both as judged by mid-afternoon shadows depicted on the satellite imagery), and that had more than three rooms, space separating them from nearby structures, electricity and a thick tree canopy.
Only three structures fit the criteria. The buildings also appeared to be the best fortified and among the largest in Parachinar. Two are clearly residences, the study states. The third may be a prison. But whatever the third structure is, it has “one of the best maintained gardens in all of Parachinar,” the study says.
While the three structures meet all six of the criteria that the researchers believe would be required for lodging bin Laden, an additional 16 structures in Parachinar appear to meet five of the six criteria. If bin Laden is not in the first three structures, the U.S. military should investigate these other buildings, the study urges.