In the late summer of 1893, following the Congress of Mathematicians held in
Chicago, Felix Klein gave two weeks of lectures on the current state of
mathematics. Rather than offering a universal perspective, Klein presented his
personal view of the most important topics of the time. It is remarkable how
most of the topics continue to be important today. Originally published in 1893
and reissued by the AMS in 1911, we are pleased to bring this work into
print once more with this new edition.
Klein begins by highlighting the works of Clebsch and of Lie. In particular,
he discusses Clebsch's work on Abelian functions and compares his approach to
the theory with Riemann's more geometrical point of view. Klein devotes two
lectures to Sophus Lie, focussing on his contributions to geometry, including
sphere geometry and contact geometry.
Klein's ability to connect different mathematical disciplines clearly comes
through in his lectures on mathematical developments. For instance, he
discusses recent progress in non-Euclidean geometry by emphasizing the
connections to projective geometry and the role of transformation groups. In
his descriptions of analytic function theory and of recent work in
hyperelliptic and Abelian functions, Klein is guided by Riemann's geometric
point of view. He discusses Galois theory and solutions of algebraic equations
of degree five or higher by reducing them to normal forms that might be solved
by non-algebraic means. Thus, as discovered by Hermite and Kronecker, the
quintic can be solved "by elliptic functions". This also leads to Klein's
well-known work connecting the quintic to the group of the icosahedron.
Klein expounds on the roles of intuition and logical thinking in
mathematics. He reflects on the influence of physics and the physical world on
mathematics and, conversely, on the influence of mathematics on physics and the
other natural sciences. The discussion is strikingly similar to today's
discussions about “physical mathematics”.
There are a few other topics covered in the lectures which are somewhat removed
from Klein's own work. For example, he discusses Hilbert's proof of the
transcendence of certain types of numbers (including $\pi$ and $e$), which
Klein finds much simpler than the methods used by Lindemann to show the
transcendence of $\pi$. Also, Klein uses the example of quadratic forms (and
forms of higher degree) to explain the need for a theory of ideals as developed
by Kummer.
Klein's look at mathematics at the end of the 19th Century remains compelling
today, both as history and as mathematics. It is delightful and fascinating to
observe from a one-hundred year retrospect, the musings of one of the masters
of an earlier era.
Readership
Graduate students, research mathematicians, and mathematical
historians.