Synthetic Philosophy of Contemporary Mathematics

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Synthetic Philosophy of Contemporary Mathematics
Fernando Zalamea

Translated by Zachary Luke Fraser

Synthetic Philosophy of Contemporary Mathematics is a unique and unprecedented book, and a much needed one. Fernando Zalamea (Professor of Mathematics at the Universidad Nacional de Colombia) offers a synthetic perspective on the vast spectrum of contemporary mathematics, together with an analysis of the new philosophical problems originating therein. 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.

The first part of the text discusses the specificity of modern (1830-1950) and contemporary (1950 to the present) mathematics, and offers an extensive review of how philosophy of mathematics addressed it (or failed to). 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. Drawing on these concrete examples, the third part proposes some generic outlines for synthesis.

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.

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, NY

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

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

Part One: The General Environment of Contemporary Mathematics
Specificity of Modern and Contemporary Mathematics
Advanced Mathematics in Treatises on Mathematical Philosophy: A Bibliographical Report
Towards a Synthetic Philosophy of Contemporary Mathematics

Part Two: Case Studies
Grothendieck: Forms of High Mathematical Creativity
Eidal Mathematics: Serres, Langlands, Lawvere, Shelah
Quiddital Mathematics: Atiyah, Lax, Connes, Kontsevich
Archeal Mathematics: Freyd, Simpson, Zilber, Gromov

Part Three: Sketches of Synthesis
Fragments of a Transitory Ontology
Comparative Epistemology and Sheaving
Phenomenology of Mathematical Creativity
Mathematics and Cultural Circulation

Synthetic Philosophy of Contemporary Mathematics
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