Book Review

That’s right, I do book reviews.

Bernhard Riemann

Turning Points In the Conception of Mathematics

Bernhard Riemann has been one of my favorite mathematicians ever since I was a wee lad reading ET Bell’s Men of Mathematics. He had a humility that was completely at odds with his genius and creativity. There have been mathematicians that were more prolific, but I don’t know of another that was more creative. As an indication of his creativity consider this. Virtually every discovery of Gauss that wasn’t published immediately (e.g. in the Disquisitiones Arithmeticae) and even some that he was given credit for (e.g. the fundamental theorem of algebra) were discovered within about 30 years of the time he discovered them. This applies to the least squares, non-Euclidean Geometry, properties of the elliptic integrals, the list goes on and on. On the other hand, some of the discoveries of Riemann weren’t understood for fifty years after he published. To me this indicates that Gauss was a mathematician who was like any other mathematician, just much, much smarter, while Riemann was something different altogether.

Other than Riemanns’ mathematics, his life was not extraordinary. He was born in 1826 in Germany, educated and taught in Germany most of his life, with some short breaks to recover in Italy.

I enjoyed this book. It didn’t have the excitement that ET Bells book has, but it covered the math much more thoroughly. Riemanns view of complex variables is explored extensively. I had wondered since my days in complex variables class, why are they called the Cauchy-Riemann equations? If Cauchy had already discovered them, what did Riemann do? A lot, as it turns out. The important point to emphasize is that Riemann viewed complex functions holistically, rather than as a given (power series) representation. His investigation of the hypergeometric function was a clear break from the previous methods of analyzing that function. Riemanns investigations into real functions paved the way for Cantors investigations into set theory. Cantors story is a bit more exciting than Riemanns real variables investigations, but it was interesting backstory. It was interesting to learn that Riemanns focus on pointwise convergence, rather than uniform may have pushed mathematics in the wrong direction for a bit.

In addition to these topics, Riemanns physics investigations, his geometry, and his famous number theory paper are all explored, as well as his general philosophy. I liked the extensive quotes given from the period, especially the German original. It’s such a beautiful language.

In short, I would recommend this book to anyone who is obsessed with mathematics, and Riemann in particular


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