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**Axiomatic method and category theory.**
*(English)*
Zbl 1284.03007

Synthese Library 364. Cham: Springer (ISBN 978-3-319-00403-7/hbk; 978-3-319-00404-4/ebook). xi, 285 p. (2014).

The monograph discusses the philosophy of axiomatic methods in 20th century mathematics. Focusing on Hilbert’s method and its derivations, the author analyses the foundation, the influences and the consequences of the axiomatic method(s) in the development and the understanding of mathematical thought.

The author defends the thesis that mathematics is ultimately based on the empirical world, and thus criticises Hilbert’s and the axiomatic view since it leads to an essentially metaphysical position. The author proposes a different method, alternative to the axiomatic approach, based and justified within an interpretation of category theory, which comes from Lawvere’s approach to logic, and that is, in a sense supported by the foundation of homotopy type theory.

In many aspects, the book is controversial: some arguments may not be convincing; in places, the author seems to have a non-standard understanding of some parts of the current research in mathematical logic; some positions expressed in the text are deviations from the mainstream among mathematicians. All these facts may be disturbing for the reader. Nevertheless, the reader of this work, even if not supporting or accepting the author’s thesis, will find an in-depth, interesting and inspiring analysis of the philosophy behind the axiomatic description of mathematics, and, for sure, the monograph provides material for serious thoughts about mathematical methodology, not simply giving for granted the usual formal presentation which became standard after Hilbert.

In summary, the book is divided into three parts. Part one critically exposes and analyses the main milestones in the development and establishment of the axiomatic method: Euclid, Hilbert, Bourbaki, and the categorical approach, represented by Lawvere. Relations to the great tradition of philosophy are made explicit, and many hidden assumptions and non-evident consequences about the axiomatic systems are presented and discussed in depth. Part two focuses on the central problem of identity of mathematical objects. Critical aspects in the concept of equality, and on the fundamental nature of mathematical entities are illustrated and analysed within different variants of the axiomatic methods, showing a number of hidden pitfalls. Part three prepares and exposes the author’s proposal for a reform of the axiomatic method in order to solve the philosophical difficulties illustrated in the rest of the book.

The author defends the thesis that mathematics is ultimately based on the empirical world, and thus criticises Hilbert’s and the axiomatic view since it leads to an essentially metaphysical position. The author proposes a different method, alternative to the axiomatic approach, based and justified within an interpretation of category theory, which comes from Lawvere’s approach to logic, and that is, in a sense supported by the foundation of homotopy type theory.

In many aspects, the book is controversial: some arguments may not be convincing; in places, the author seems to have a non-standard understanding of some parts of the current research in mathematical logic; some positions expressed in the text are deviations from the mainstream among mathematicians. All these facts may be disturbing for the reader. Nevertheless, the reader of this work, even if not supporting or accepting the author’s thesis, will find an in-depth, interesting and inspiring analysis of the philosophy behind the axiomatic description of mathematics, and, for sure, the monograph provides material for serious thoughts about mathematical methodology, not simply giving for granted the usual formal presentation which became standard after Hilbert.

In summary, the book is divided into three parts. Part one critically exposes and analyses the main milestones in the development and establishment of the axiomatic method: Euclid, Hilbert, Bourbaki, and the categorical approach, represented by Lawvere. Relations to the great tradition of philosophy are made explicit, and many hidden assumptions and non-evident consequences about the axiomatic systems are presented and discussed in depth. Part two focuses on the central problem of identity of mathematical objects. Critical aspects in the concept of equality, and on the fundamental nature of mathematical entities are illustrated and analysed within different variants of the axiomatic methods, showing a number of hidden pitfalls. Part three prepares and exposes the author’s proposal for a reform of the axiomatic method in order to solve the philosophical difficulties illustrated in the rest of the book.

Reviewer: Marco Benini (Buccinasco)

### MSC:

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

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

00A30 | Philosophy of mathematics |

00A35 | Methodology of mathematics |

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