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**Thinking in structures and its history. About the power of mathematical proof.
(Denken in Strukturen und seine Geschichte. Von der Kraft des mathematischen Beweises.)**
*(German)*
Zbl 1419.00006

Berlin: Springer (ISBN 978-3-662-56376-2/pbk; 978-3-662-56377-9/ebook). xii, 383 p. (2018).

This is a comprehensive, elementary historical and systematic introduction to the philosophy of mathematics written for a broader audience of non-specialists. The author’s basic assumption is the understanding of science as a structured system of thoughts, not as a collection of more or less accepted propositions. All scientific propositions are standing in logical relations to each other and can be deduced from certain basic principles (p.V).

The author starts with reflections on thinking and storytelling, arriving at the problems of ordinary language and logical contradictions: antinomies, paradoxes, and aporias (Chapter 2). He deals with the concept of infinity in its mythical and ancient origins (Chapter 3). He sees the beginnings of thinking in structures in ancient sources, in particular in Euclid’s Elements (Chapter 4), but considers also influences from other parts of the world (Chapter 5). For the author, sets, relations and numbers are the basic concepts of structural thinking (Chapter 6). Its further development in the Renaissance is seen in Descartes’ analytical geometry, the mathematization of science (Galileo, Newton), and the theory of infinitesimals in Leibniz and Newton (Chapter 7). Modern mathematics is characterized as being organized in axiomatic-deductive systems following David Hilbert’s revision of Euclid’s model (Chapter 8). ZFC is presented as the foundational theory for mathematics (Chapter 9). Set theory itself is based on formal language, i.e., symbolic logic with its proof methods and their limits indicated by K.Gödel’s incompleteness theorems (Chapter 10). The author sees important applications of formal languages and structural thinking in artificial intelligence, e.g., in neural networks, and machine learning, supporting conceptions of action and decision theory, e.g., in deontic contexts (Chapter 11). In the last chapter (Chapter 12) the author discusses several alternative philosophical accounts, e.g. structural realism and constructivism. He considers the role of reasoning in storytelling, criticizes neo-positivistic approaches, structuralism and post-modernism. He shares C.McGinn’s “transcendental naturalism” (p.364).

This elementary and readable introduction combines a vast range of topics motivating interested readers to dig deeper into the field of foundations.

The author starts with reflections on thinking and storytelling, arriving at the problems of ordinary language and logical contradictions: antinomies, paradoxes, and aporias (Chapter 2). He deals with the concept of infinity in its mythical and ancient origins (Chapter 3). He sees the beginnings of thinking in structures in ancient sources, in particular in Euclid’s Elements (Chapter 4), but considers also influences from other parts of the world (Chapter 5). For the author, sets, relations and numbers are the basic concepts of structural thinking (Chapter 6). Its further development in the Renaissance is seen in Descartes’ analytical geometry, the mathematization of science (Galileo, Newton), and the theory of infinitesimals in Leibniz and Newton (Chapter 7). Modern mathematics is characterized as being organized in axiomatic-deductive systems following David Hilbert’s revision of Euclid’s model (Chapter 8). ZFC is presented as the foundational theory for mathematics (Chapter 9). Set theory itself is based on formal language, i.e., symbolic logic with its proof methods and their limits indicated by K.Gödel’s incompleteness theorems (Chapter 10). The author sees important applications of formal languages and structural thinking in artificial intelligence, e.g., in neural networks, and machine learning, supporting conceptions of action and decision theory, e.g., in deontic contexts (Chapter 11). In the last chapter (Chapter 12) the author discusses several alternative philosophical accounts, e.g. structural realism and constructivism. He considers the role of reasoning in storytelling, criticizes neo-positivistic approaches, structuralism and post-modernism. He shares C.McGinn’s “transcendental naturalism” (p.364).

This elementary and readable introduction combines a vast range of topics motivating interested readers to dig deeper into the field of foundations.

Reviewer: Volker Peckhaus (Paderborn)