Alain Connes Books

Do numbers and mathematical objects exist independently of human minds, or are they inventions of our cognition? Are they discovered, as Plato suggested, or constructed by us? Is mathematics a universal language capable of facilitating communication with extraterrestrial civilizations, or is it merely a product of human evolution? Do physical laws genuinely adhere to mathematical principles, or do they merely appear to do so because physicists have learned to interpret them mathematically? These fundamental questions create a divide between Jean-Pierre Changeux, a renowned neurobiologist, and Alain Connes, a prominent mathematician. Their discussions explore the implications of mathematical objects on brain organization and function, examining how brain development influences mathematical reasoning and intelligence. They also delve into ethical considerations, questioning the neural foundations of morality and its social expressions. This engaging dialogue reflects a profound disagreement while seeking mutual understanding, echoing the inquiries of Poincaré, Hadamard, and von Neumann about human experience and intellectual limits. Central to their exploration is the inquiry into why order exists in the world and why it is comprehensible to humanity.

Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.