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**Kant’s mathematical world. Mathematics, cognition, and experience.**
*(English)*
Zbl 1537.01003

Cambridge: Cambridge University Press (ISBN 978-1-108-45510-7/pbk; 978-1-108-42996-2/hbk; 978-1-108-55574-6/ebook). xiv, 302 p. (2022).

The book deals with the role of mathematics in Immanuel Kant’s trancendental account of the reform of metaphysics, in particular in his Critique of pure reason. The author’s reconstruction focuses on intuition in mathematics and the representation of magnitudes. His book is “an attempt to describe Kant’s theory of magnitudes and its foundational role in Kant’s account of mathematical cognition and our cognition of the world” (p.xi). For Kant “mathematics is the science of magnitudes, and the world of experience is a world of magnitudes”, so that “the world is fundamentally mathematical in character” (p.1). Within the theory of magnitudes the distinction between quantum and quantitas is essential, a quantum being understood as a homogeneous manifold in intuition, whereas the concept of quantitas allows segments of space and time to be determined.

The material is divided into two parts, separated by an interlude. Part I explores mathematics and magnitudes in Kant’s transcendental programme. The author presents space, time and mathematics in the Critique of pure reason (Chapter 2), relates magnitudes, mathematics and experience to the axioms of intuition (Chapter 3), discusses the conceptual connection between magnitudes and continuity (Chapter 4), and finally presents singularity, continuity and concreteness as variations of representation.

The interlude contextualises Kant’s philosophy of mathematics by developing the theory of magnitudes in Euclid and the Euclidean mathematical tradition as the basis for Kant’s critical revisions.

Part II reconstructs Kant’s theory of magnitudes, intuition and measurement. Chapter 7 deals with the concept of homogeneity, Chapter 8 explores its implications for Kant’s new metaphysics of quantity, which differs significantly from the theories of his predecessors Leibniz, Wolff and Baumgarten. Chapter 9 aims to show how Kant’s theory of magnitudes helps to explain mathematical cognition and the mathematical character of experience. With the help of the concepts of measurement and equality Kant’s mereological theory of magnitudes can be used as the basis for mathematics. In the final Chapter 10, the author recalls his aim “to fundamentally transform our current understanding of both Kant’s philosophy of mathematics and his account of our understanding of that theory” (p.281), which cannot be explained on the basis of the late 19th-century paradigm of the arithmetization of mathematics.

The material is divided into two parts, separated by an interlude. Part I explores mathematics and magnitudes in Kant’s transcendental programme. The author presents space, time and mathematics in the Critique of pure reason (Chapter 2), relates magnitudes, mathematics and experience to the axioms of intuition (Chapter 3), discusses the conceptual connection between magnitudes and continuity (Chapter 4), and finally presents singularity, continuity and concreteness as variations of representation.

The interlude contextualises Kant’s philosophy of mathematics by developing the theory of magnitudes in Euclid and the Euclidean mathematical tradition as the basis for Kant’s critical revisions.

Part II reconstructs Kant’s theory of magnitudes, intuition and measurement. Chapter 7 deals with the concept of homogeneity, Chapter 8 explores its implications for Kant’s new metaphysics of quantity, which differs significantly from the theories of his predecessors Leibniz, Wolff and Baumgarten. Chapter 9 aims to show how Kant’s theory of magnitudes helps to explain mathematical cognition and the mathematical character of experience. With the help of the concepts of measurement and equality Kant’s mereological theory of magnitudes can be used as the basis for mathematics. In the final Chapter 10, the author recalls his aim “to fundamentally transform our current understanding of both Kant’s philosophy of mathematics and his account of our understanding of that theory” (p.281), which cannot be explained on the basis of the late 19th-century paradigm of the arithmetization of mathematics.

Reviewer: Volker Peckhaus (Paderborn)

### MSC:

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |

01A50 | History of mathematics in the 18th century |

00A30 | Philosophy of mathematics |