A panoramic survey of the vast spectrum of modern and contemporary mathematics and the new philosophical possibilities they suggest.

A panoramic survey of the vast spectrum of modern and contemporary mathematics and the new philosophical possibilities they suggest, this book gives the inquisitive non-specialist an insight into the conceptual transformations and intellectual orientations of modern and contemporary mathematics.

The predominant analytic approach, with its focus on the formal, the elementary and the foundational, has effectively divorced philosophy from the real practice of mathematics and the profound conceptual shifts in the discipline over the last century. The first part discusses the specificity of modern (1830–1950) and contemporary (1950 to the present) mathematics, and reviews the failure of mainstream philosophy of mathematics to address this specificity. Building on the work of the few exceptional thinkers to have engaged with the “real mathematics” of their era (including Lautman, Deleuze, Badiou, de Lorenzo and Châtelet), Zalamea challenges philosophy's self-imposed ignorance of the “making of mathematics.”

In the second part, thirteen detailed case studies examine the greatest creators in the field, mapping the central advances accomplished in mathematics over the last half-century, exploring in vivid detail the characteristic creative gestures of modern master Grothendieck and contemporary creators including Lawvere, Shelah, Connes, and Freyd.

Drawing on these concrete examples, and oriented by a unique philosophical constellation (Peirce, Lautman, Merleau-Ponty), in the third part Zalamea sets out the program for a sophisticated new epistemology, one that will avail itself of the powerful conceptual instruments forged by the mathematical mind, but which have until now remained largely neglected by philosophers.

Fernando Zalamea is Professor of Mathematics at the Universidad Nacional de Colombia.

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This book, elegantly and clearly translated by Z.L. Fraser provides, for the Anglophone world, insight into Zalamea's renewal of philosophy through mathematical (and synthetical) means.—Tzuchien Tho, MUTE—

Zalamea's book is thematically vast. It is truly astounding to behold the rich range of mathematical themes that are touched upon, arguably including all of the most important objects of contemporary exploration.

—Giuseppe Longo, Speculations—

This is a weighty and daring book. It proposes a new philosophy of mathematics, based on a detailed knowledge of the most recent work in advanced mathematics, and constructed in explicit contrast with the traditional analytical approach … this new synthetic and open-minded approach is no doubt worthy of attention, and philosophers who dare to make an effort will surely reap the reward.

—Paloma Pérez-Ilzarbe, MathSciNet—