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**Diagrammatic immanence. Category theory and philosophy.**
*(English)*
Zbl 1367.03002

Edinburgh: Edinburgh University Press (ISBN 978-1-4744-0417-4/hbk; 978-1-4744-0418-1/ebook). vii, 256 p. (2016).

This philosophical essay focuses on the concept of immanence, and the thesis it aims to is demonstrating that category theory is the adequate mathematical tool to investigate immanence immanently.

The present review emphasizes the mathematically relevant aspects of the book, while the reader interested in the proper philosophical content is referred to the excellent review by Jean-Pierre Marquis, http://ndpr.nd.edu/news/diagrammatic-immanence-category-theory-and-philosophy/ (URL consulted on June 29th, 2017).

The text is divided into six chapters: the first one is devoted to analyse Spinoza’s ethics; the third discusses about Pierce’s semiotics; the fifth presents Deleuze’s approach to the logic of sense. The even chapters illustrate the fundamentals of category theory according to the philosophical lines preceding them: Chapter 2 introduces categories and functors; Chapter 4 speaks about functor categories and presheaves; Chapter 6 is about adjunctions and elementary toposes.

This organisation, alternating philosophy and mathematics, tries to mimic the structure of Spinoza’s ethics, so it should be understood as part of the content itself, a fact clarified in the first chapter.

The philosophical chapters vividly expose the author’s interpretation of the discussed philosophers’ thought. However, despite the efforts of the author, some passages are hard for a reader not accustomed to the philosophical way of exposition, and, being the text an essay, previous knowledge of the works and ideas of the illustrated philosophers is needed to understand.

The mathematical chapters are shorter, descriptive and tailored toward a public of philosophers/humanists rather than mathematicians/scientists. With this proviso in mind, they succeed in conveying the idea of what categories are and how the fundamental ideas behind them work, although a mathematician would object on some lack of formal rigor or about the absence of proper formalism. This is not a failure in the exposition since the objective is not to teach category theory, but to show how it provides a concrete instance/tool to immanent reasoning. In this respect, the exposition is fully satisfactory as soon as the reader accepts, at least provisionally, the author’s thesis.

The overall picture clearly shows that this is a philosophical essay whose intended public is composed by philosophers. Nevertheless, for a mathematician is may be of interest because it provides a different, alternative, and in some way provocative view of category theory. In turn, this view suggests hints toward an immanent foundation of mathematics although the text does not elaborate this line specifically. It is a stimulating way of thinking out of the usual tracks of mathematics, and it could bring new ideas and insights, or, perhaps, just provide a critical, non-standard point of view based on a solid philosophical tradition.

The present review emphasizes the mathematically relevant aspects of the book, while the reader interested in the proper philosophical content is referred to the excellent review by Jean-Pierre Marquis, http://ndpr.nd.edu/news/diagrammatic-immanence-category-theory-and-philosophy/ (URL consulted on June 29th, 2017).

The text is divided into six chapters: the first one is devoted to analyse Spinoza’s ethics; the third discusses about Pierce’s semiotics; the fifth presents Deleuze’s approach to the logic of sense. The even chapters illustrate the fundamentals of category theory according to the philosophical lines preceding them: Chapter 2 introduces categories and functors; Chapter 4 speaks about functor categories and presheaves; Chapter 6 is about adjunctions and elementary toposes.

This organisation, alternating philosophy and mathematics, tries to mimic the structure of Spinoza’s ethics, so it should be understood as part of the content itself, a fact clarified in the first chapter.

The philosophical chapters vividly expose the author’s interpretation of the discussed philosophers’ thought. However, despite the efforts of the author, some passages are hard for a reader not accustomed to the philosophical way of exposition, and, being the text an essay, previous knowledge of the works and ideas of the illustrated philosophers is needed to understand.

The mathematical chapters are shorter, descriptive and tailored toward a public of philosophers/humanists rather than mathematicians/scientists. With this proviso in mind, they succeed in conveying the idea of what categories are and how the fundamental ideas behind them work, although a mathematician would object on some lack of formal rigor or about the absence of proper formalism. This is not a failure in the exposition since the objective is not to teach category theory, but to show how it provides a concrete instance/tool to immanent reasoning. In this respect, the exposition is fully satisfactory as soon as the reader accepts, at least provisionally, the author’s thesis.

The overall picture clearly shows that this is a philosophical essay whose intended public is composed by philosophers. Nevertheless, for a mathematician is may be of interest because it provides a different, alternative, and in some way provocative view of category theory. In turn, this view suggests hints toward an immanent foundation of mathematics although the text does not elaborate this line specifically. It is a stimulating way of thinking out of the usual tracks of mathematics, and it could bring new ideas and insights, or, perhaps, just provide a critical, non-standard point of view based on a solid philosophical tradition.

Reviewer: Marco Benini (Buccinasco)

### MSC:

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

00A30 | Philosophy of mathematics |

18-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory |

03A05 | Philosophical and critical aspects of logic and foundations |

18A15 | Foundations, relations to logic and deductive systems |