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Information for current students:
Information
for prospective students:
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Math 497 Spring '07 |
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Instructor: Prof. Stephen
Monk |
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Course
description and expectations:
Although it is one of the most important ideas in calculus, the concept of limit remains an extremely elusive one for most students. Few have any sense of the role it plays in the subject, the questions it helps us address, or the broader mathematical ideas that lie behind it. Because of the way limits are treated in elementary calculus courses, students tend to see the concept as part of the "theory" of calculus, of interest to mathematicians, and having little to do with the way they use or understand calculus themselves. The
concept of limit is one of the great mathematical achievements of all
time. It is the capstone of an intellectual struggle, going back to
the Ancient Greeks, in which mathematicians, scientists, and philosophers
have addressed the problems and paradoxes of infinite processes: How
can we add an infinite set of numbers? How can we measure the area of
a region with a curved boundary, if measuring area means the counting
the number of unit squares that fit into a region? How can we talk about
rate of change at an instant, if, by definition, change has to occur
over a period of positive duration? The Greeks made important contributions
to the solution of some of these problems. But even the great Newton
and Leibniz, the inventors of calculus, failed to solve these problems
in a way that was completely satisfactory to their contemporaries or
able to withstand the test of time. It was not until the second half
of the 19th century that mathematicians were able to completely and
rigorously solve these problems. Doing so required that they rethink
and rework many of the main ideas of mathematics, including the concept
of function and our system of real numbers. Although satisfying to mathematicians,
this solution has the unfortunate quality of being completely removed
from common-sense notions of area, rate of change, and number, as well
as from ordinary ways of thinking and speaking. After 2,500 years of
confusion and error, they were willing to pay a very high price in intuitive
appeal in order to get things right!! In this course,
we will go back to the underlying difficulties involved in analyzing
and understanding infinite processes and explore how the Greeks dealt
with them. We will study approaches to solving these problems developed
between the 15th and 18th centuries, as a way of grappling further with
these issues, as well as to make connections between the way one normally
thinks about concepts like area, rate of change, and number, and the
highly formalized and abstract concept of limit we see in calculus textbooks
today. Although this course
will draw heavily on historical material, it is not a systematic course
in the history of calculus. A wide variety of examples of mathematical
problems, many of them historical, will be discussed. These will be
used as to investigate the problems inherent in the analysis of infinite
processes and to develop careful and rigorous ways to deal with them.
The goal is to slowly build an understanding of infinite processes from
which the particular approach to describing them developed in the 19th
century can make sense. The work of this
course will consist of in-class discussions based on homework assignments
that involve reading about the underlying mathematical ideas, solving
mathematics problems, and reflecting on classroom discussions. Students
will write two brief papers on these mathematical ideas. There will
be no examinations. Students will be expected to be familiar with first-year college calculus. But more important, they should be willing to think about and discuss a variety of approaches to mathematical concepts. If
you would like more information about this course, just click on names
of documents under Information for prospective students, or contact
Steve Monk. monk@math.washington.edu. |
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