Synthetic Philosophy of Contemporary Mathematics

Fernando Zalamea

December 2012

Translated by Zachary Luke Fraser

Paperback 115x175mm, 380pp

ISBN 978-0-9567750-1-6

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*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, American Mathematical Society's MathSciNet

*Zalamea is clearly on the cutting edge of theorizing potential
intersections between networks, math, and philosophy. Few thinkers are able to bring together insights from as diverse fields in such as exciting manner as Zalamea.* - Christopher Vitale
(Assistant Professor, Media Studies, Pratt Institute, New York)

*With high professional competence in mathematics and philosophy, and writing in masterful prose, Zalamea opens up a breathtaking insight into advanced contemporary mathematics, enlightening its magical power with the powerful paradigm of gestural dynamics as developed by Valéry, Merleau-Ponty, and Châtelet.* - Guerino Mazzola (Author of *The Topos of Music*, Professor of Mathematical Music Theory and Creativity, School of Music, University of Minnesota)

*From a global perspective, Zalamea's book is a defining moment in the enduring legacy of Universalist knowledge, comparable in its historical significance to the works of Oresme, Peirce and Lautman. It outlines contemporary mathematics as a germinal edifice that, far from restricting or moderating various local fields of knowledge, broadens and deepens the scope of synthesis and fusion between them, laying out the foundations of a rational knowledge as a transit system endowed with universal orientation - a plastic, bottomless-up protean meshwork where mathematics is free from a fixed framework, and synthetic philosophy is philosophy in the making.* - Reza Negarestani

In
*Synthetic Philosophy of Contemporary Mathematics*, Fernando Zalamea (Professor of Mathematics, Universidad Nacional de Colombia) offers the reader a panoramic view of the vast spectrum of modern and contemporary mathematics, and the new philosophical possibilities they suggest.

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 of Zalamea's text 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.

Examining the few exceptional thinkers to have engaged with the 'real mathematics' of their era (including Lautman, Deleuze, Badiou, de Lorenzo and Châtelet), Zalamea seeks to expand and intensify this engagement, challenging 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, compiling a map of the central advances accomplished in mathematics over the last half-century. In these case studies the characteristic creative gestures of modern master Grothendieck and contemporary creators including Lawvere, Shelah, Connes and Freyd are explored in vivid detail.

Drawing on these concrete examples, the third part proposes some generic outlines for synthesis. Oriented by a unique philosophical constellation (Peirce, Lautman, Merleau-Ponty), Zalamea uses this survey of mathematical experience to set 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.

*Synthetic Philosophy of Contemporary Mathematics* is a unique and unprecedented book, and a much needed one. The book makes available to the inquisitive non-specialist the conceptual transformations and intellectual orientations of modern and contemporary mathematics, and their significance for speculative philosophical thought.

Zalamea's book serves as a conceptual introduction to mathematical themes rarely discussed outside specialist circles, and as a critical lens by means of which today's mathematics may aid us in the configuration of new cultural perspectives.

If analytic philosophy was forged in the fires of set theory and classical logic at the beginning of the twentieth century, then today, at the dawn of the twenty-first, and around the scaffolding of category theory and the logic of sheaves, it is time for a complementary, synthetic philosophy to be built.

Contents

Introduction: Traditional Options for the Philosophy of Mathematics and Prospectus for this Essay

*Part One: The General Environment of Contemporary Mathematics*

1. Specificity of Modern and Contemporary Mathematics

2. Advanced Mathematics in Treatises on Mathematical Philosophy: A Bibliographical Report

3. Towards a Synthetic Philosophy of Contemporary Mathematics

*Part Two: Case Studies*

4. Grothendieck: Forms of High Mathematical Creativity

5. Eidal Mathematics: Serres, Langlands, Lawvere, Shelah

6. Quiddital Mathematics: Atiyah, Lax, Connes, Kontsevich

7. Archeal Mathematics: Freyd, Simpson, Zilber, Gromov

*Part Three: Sketches of Synthesis*

8. Fragments of a Transitory Ontology

9. Comparative Epistemology and Sheaving

10. Phenomenology of Mathematical Creativity

11. Mathematics and Cultural Circulation

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