AP English A Exam Final Page 9
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Questions 2639. Read the following passage carefully before you choose your answers.
( The following passage is from a contemporary biography about a mathematician. )
For [Paul] Erdös, mathematics was a glorious combination of science and art. On the one hand, it was the science of certainty, because its conclusions were logically unassailable. Unlike biologists, chemists, or even physicists, Erdös, Graham, and their fellow mathematicians prove things. Their conclusions follow syllogistically from premises, in the same way that the conclusion “Bill Clinton is mortal" follows from the premises “All presidents are mortal” and “Bill Clinton is a president.” On the other hand, mathematics has an aesthetic side.
A conjecture can be “obvious” or “unexpected.” A result can be “trivial" or “beautiful.” A proof can be “messy,” “surprising,” or, as Erdös would say, “straight from the Book." In a good proof, wrote Hardy, “there is a very high degree of imexpectedness, combined with inevitability and economy. The argument takes so odd and surprising a form; the weapons used seem so childishly simple when compared with the farreaching consequences; but there is no escape from the conclusions."
What is more, a proof should ideally provide insight into why a particular result is true. Consider one of the most famous results in modern mathematics, the Four Color Map Theorem, which states that no more than four colors are needed to paint any conceivable flat map of real or imaginary countries in such a way that no two bordering countries have the same color. From the middle of the nineteenth century, most mathematicians believed that this seductively simple theorem was true, but for 124 years a parade of distinguished mathematicians and dedicated amateurs searched in vain for a proofand a few contrarians looked for a counterexample. “When I started at AT&T,” said Graham, “there was a mathematician there named E. F. Moore who was convinced that he
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Questions 2639. Read the following passage carefully before you choose your answers.
( The following passage is from a contemporary biography about a mathematician. )
For [Paul] Erdös, mathematics was a glorious combination of science and art. On the one hand, it was the science of certainty, because its conclusions were logically unassailable. Unlike biologists, chemists, or even physicists, Erdös, Graham, and their fellow mathematicians prove things. Their conclusions follow syllogistically from premises, in the same way that the conclusion “Bill Clinton is mortal" follows from the premises “All presidents are mortal” and “Bill Clinton is a president.” On the other hand, mathematics has an aesthetic side.
A conjecture can be “obvious” or “unexpected.” A result can be “trivial" or “beautiful.” A proof can be “messy,” “surprising,” or, as Erdös would say, “straight from the Book." In a good proof, wrote Hardy, “there is a very high degree of imexpectedness, combined with inevitability and economy. The argument takes so odd and surprising a form; the weapons used seem so childishly simple when compared with the farreaching consequences; but there is no escape from the conclusions."
What is more, a proof should ideally provide insight into why a particular result is true. Consider one of the most famous results in modern mathematics, the Four Color Map Theorem, which states that no more than four colors are needed to paint any conceivable flat map of real or imaginary countries in such a way that no two bordering countries have the same color. From the middle of the nineteenth century, most mathematicians believed that this seductively simple theorem was true, but for 124 years a parade of distinguished mathematicians and dedicated amateurs searched in vain for a proofand a few contrarians looked for a counterexample. “When I started at AT&T,” said Graham, “there was a mathematician there named E. F. Moore who was convinced that he