When Bernhard Riemann had to give a lecture as part of his Habilitation examination, he proposed three topics. The third was "On the Hypotheses that Lie at the Foundations of Geometry." C. F.Gauss, who was the main examiner, chose that topic as the one Riemann should talk about. The result was one of the most famous texts in the history of geometry. In a short lecture, using almost no equations, Riemann sketched a whole new approach to geometry. One can find there the roots of the theory of n-dimensional manifolds, of Riemannian geometry and of General Relativity.
It took a while to sort it all out, of course. The "sorting out" actually seems to have taken two quite different routes. The first was the more technical one of building the mathematics that Riemann only hinted at. The second was more philosophical, trying to determine how our ideas of space emerged, the relation between physical space and the various mathematically plausible geometries, and the implications for physical science. It is the latter thread that Peter Pesic follows in this very useful collection of papers.
Pesic recently edited a new edition, also for Dover, of Gauss's famous 1827 memoir on curved surfaces. This collection continues the story. It opens with Riemann's Habilitation lecture, from 1854, and follows that with a few selections from Riemann's papers. Then come essays by Helmholtz, Clifford, Newcomb, Poincaré, Klein, (Élie) Cartan, and Einstein (who is represented by several articles). The most recent of these is from 1934. Several of the texts have been newly translated, and Pesic has added an informative introduction and extensive notes.
As the emphasis on Einstein suggests, the essays collected here, while mathematically well informed, are more interested in philosphy and physics. One of the big questions addressed is the geometry of the real world in which we live and how we might figure out more about it. The essays are also mostly non-technical. They are not "easy to read," however! Some very subtle ideas are discussed here.
Consider, for example, the following: in our everyday experience of physical space, we find that it is possible to move rigid objects around without changing their shape. Surely this imposes restrictions on the geometry of space. What are they?
My one caveat is with the title of the book. If anything, these essays emphasize the crucial role of geometry in our understanding of the world. Rather than leading "beyond" it, they lead us far more deeply into it. Some of them even toy with the idea that geometry is the only ultimate reality.
This collection will provide food for thought for both mathematicians and those interested in the relationship between mathematics and our understanding of reality. The largely non-technical nature of the essays makes them widely accessible. The book would make a wonderful text for a senior seminar.
This one is a winner. Pesic has put us all in his debt.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.