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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

A Radical Approach to Real Analysis

Chapter 1. Crisis in Mathematics: Fourier's Series

The 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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