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> Chapter 1: Crises in Mathematics: Fourier's Series
Historical Background
Fitting the Initial Condition
The General Solution
Obstacles
Term-by-term Integration
Chapter 2: Infinite Summations
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A Radical Approach to Real Analysis
Chapter 1. Crisis in Mathematics: Fourier's Series
T he crisis struck
four days before Christmas 1807. The edifice of calculus was shaken to
its foundations. In retrospect, the difficulties had been building for
decades. Yet while most scientists realized that something had happened,
it would take fifty years before the full impact of the event was understood.
The nineteenth century would see ever expanding investigations into the
assumptions of calculus, an inspection and refitting of the structure
from the footings to the pinnacle, so thorough a reconstruction that calculus
was given a new name: Analysis. Few of those who witnessed the
incident of 1807 would have recognized mathematics as it stood one hundred
years later. The twentieth century was to open with a redefinition of
the integral by Henri
Lebesgue and an examination of the logical underpinnings of arithmetic
by Bertrand
Russell and Alfred
North Whitehead, both direct consequences of the events set in motion
in that critical year. The crisis was precipitated by the deposition at
the Institut de France in Paris of a manuscript, Theory of the Propagation
of Heat in Solid Bodies, by the 39-year old prefect of the department
of Isère, Joseph
Fourier.
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