So Many Problems, So Little Time: Maps and Mathematics
by Jan Figa
Jan Figa is the physics/mathematics librarian at the University of Michigan.
Mapping Solutions
In the movie "A Beautiful Mind," John Nash, game theorist extraordinaire, invites his date, Alicia, to imagine any object, which he then maps using the stars of the night sky. Alicia jokingly suggests an umbrella and then an octopus. Nash, who completed his doctoral dissertation by age 21, quickly traces a few stars to yield the desired objects, thereby winning the heart of Alicia, his future wife.
Nash's Princeton classmates find his preoccupation with describing competitive foraging techniques of pigeons, even if summarized by beautiful mathematical symbols, hilarious and a waste of time. Another scene places Nash in a military installation where he displays unwavering determination and prowess by mentally sifting through thousands of arrangements of numbers to finally, after several hours, break an enemy code.
Nash's pursuits illustrate problem solving specific to mathematics. Namely, pattern recognition, abstraction into symbolic description, and finally a formalized statement of the result (known as a theorem). In a sense, the theorem embodies both the path (known as a proof) and the methods to be followed in order to arrive at a stated destination (known as a conclusion). Hence a (mathematical) proof may be considered a map demonstrating why a (mathematical) statement is true.
Mathematicians will speak of a proof, but often they will only provide a sketch of the proof with the details to be filled in by the dogged pursuer of mathematical truth. Thus, the complete proof is condensed even further to a string of references leading to the major points that contain the thrust of the proof. A proof will look like maps within a map and in this sense resembles hyperlinks within a document.
Problem solving is dynamic map making. It may be helpful to compare problem solving in mathematics to a mental discovery process that charts the possible path(s) from problem to solution. In a sense, a mathematician seeks to correctly and efficiently navigate the topography of mathematics. This exploration is highly complicated and rarely surrenders to a flowchart approach. Ability, diligence, and years of devoted study are necessary in becoming an independent wayfarer. Over time, as in the real world, new or refined tools make it possible to solve new problems or see old problems in a more revealing light. Thus, not only is the granularity of making the maps enhanced, but so is the ability to overcome mathematical obstacles. In brief, the texture of the map representing the state of mathematical knowledge is dynamic.
The very nature of mathematics, with its emphasis on problem solving, actively encourages the apprentice mathematician to learn as many kinds of theorems and types of proof as possible. The richness and connectedness of the web of mathematics become more apparent to an experienced practitioner as his or her "bag of mathematical tricks" grows in size and maturity, and specifically as a problem solver encounters a greater number of problems requiring careful examination. To foster skill and inspire ingenuity in mathematical problem solving, there are exceptional tests such as the U.S.A.'s Putnam Exam (university-level) and the International Mathematical Olympiad (high school-level). These examinations, comparable to the Olympics, celebrate mathematics and provide a competitive environment to develop a palette of problem solving skills to appreciate, attack, and to circumnavigate mathematical obstacles, if needed. These tests also serve as a reservoir of mathematically meaningful problems that may, after suitable modification, be used as a teaching tool in a case-based learning environment. This instructional approach is similar to that found in many business and law schools.
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"Mathematicians are machines which turn coffee into theorems."
Paul Erdös (1913-1996) Dr. Ernst Straus, who worked with both Albert Einstein and Erdös, wrote a tribute to Erdös which reads in part: "In our century, in which mathematics is so strongly dominated by 'theory doctors,' he has remained the prince of problem solvers and the absolute monarch of problem posers." |
"Ah ha," you will say, recognizing the way Nash probably constructed the outline of an octopus using stars. He chose stars that formed segments (resembling legs) and then pieced the segments together to form an octopus. The difficult part was to connect the segments at one star in such a way as to truly give the appearance of an octopus.
But how do you know if it is possible to make an octopus from the relative position of the stars? In mathematics, this is known as the question of existence, which if answered in the affirmative begs the "question of uniqueness" or what are the number of solutions? Some mathematicians solve problems as Field Marshall Montgomery wouldby assembling all the necessary resources before an assault. Other mathematicians solve problems as General Patton wouldby pushing until the problem surrenders. Of course, the types of mathematicians are mirrored by a generous variety of problem solving approaches.
There are two fundamental camps in mathematicspure and applied. Pure mathematics tries to artificially distinguish itself from applied mathematics by claiming not to be useful and (snobbishly) superior. In very simplistic terms, pure mathematics is inertia-laden as it seeks to build a solid foundation pushing the construction of ever more general mathematical structures. Applied mathematics seeks to create and solve suitable models that reveal the physical mechanisms for a given phenomenon (traffic flow, weather, combustion, population dynamics, etc). Interestingly, Nash's models in game theory inspired further results in diverse applied mathematical fields, such as the mathematical theory of the stock market and genetics.
Nash succeeded in solving a problem that von Neumann, a mathematical giant, could not solve, demonstrating a classical manner in which a mathematician seeks famesolving a tough problem that has stumped great minds. One such grand problem was due to Pierre de Fermat (16011665), a lawyer and amateur mathematician; it is referred to as Fermat's Last Theorem (FLT). This famous mathematical problem withstood the assault of the greatest mathematical minds for more than 350 years, until Cambridge Professor Andrew Wiles, working in secret for more than eight years, burst upon the mathematical scene in 1993 with a 150-page solution. But the "solution" contained a small flaw, which was fixed with the help of another Cambridge mathematics professor. Wiles built his solution by combining the results of others, creating new mathematical tools as needed, and receiving help from his peers with the pesky details. Wiles' approach highlights problem solving in mathematics, which is to combine, refine, and develop tools that help solve or extend results. As this process happens colleagues provide quality control.
| To learn more about Fermat's Last Theorem, visit the following Web sites:
· NOVA Online: http://www.pbs.org/wgbh/nova/proof/ (A photo of Wiles is available here)
· BBC Horizon: http://www.bbc.co.uk/science/horizon/fermat.shtml
To learn more about John Forbes Nash, visit the following Web sites:
· Biography: http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Nash.html
· Nobel Prize site: http://www.nobel.se/economics/laureates/1994/
· Autobiography: http://www.nobel.se/economics/laureates/1994/nash-autobio.html
· Resources: http://cepa.newschool.edu/het/profiles/nash.htm
· "John Nash and a Beautiful Mind," Notices of the American Mathematical Society, p 1329-1332, November 1998. |
Wiles compares solving FLT to that of an odyssey. In fact he describes his "experience of doing mathematics in terms of a journey through a dark, unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, the culmination ofand couldn't exist withoutthe many months of stumbling around in the dark that proceed them."
The conquest of FLT exemplifies the intellectual struggle of the human mind, and even made international news headlines, allowing the general populous to share in the victory. Amazingly, a musical drama, Fermat's Last Tango, based on the proof of FLT was produced by the York Theatre Company and ran Off Broadway in late 2000 and early 2001. The play brings together mathematics and theatre in an attempt to understand the human component and cultural meaning of a great intellectual achievement.
Wiles' journey describes mathematical problem solving as equivalent to exploration or possibly cartography. Explorers traverse territory, extending paths of others and sometimes retracing the steps of others. But they always create maps to enable navigation. In the beginning, these maps were not very detailed but over time they showed a textured landscape with sophisticated "pointers" that enabled even the novice searcher to retrace, flesh out, survey, and possibly extend the landscape of mathematics. The dynamic character of these maps emerges as one sees the effects of improvements in solution techniques and the accompanying results.
The problems considered by Nash and Wiles required exceptional mathematical problem solving skills. One may compare their efforts to the construction of an exotic racecar. The knowledge gained from building and testing a racer forms a map that others can study, learn and iteratively improve upon.
But not all mathematical problem solving skills and tools have to originate from exceptional problems. Calculus provides a classical example of a well-mapped branch of mathematics in which problem solving has generated the mathematical tools and results behind applications that we find indispensablethe automobile, the refrigerator, the telephone, and the computer. 
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