This book is definitely not for your garden variety, everyday mathematician. It might be for a mathematician with an interest in the philosophy of mathematics. The book has a rather long introduction (pp. ix–xiiv) by the translator, who is a friend of the author’s. The book itself is organized into chapters about concepts, logical connectors, definitions, propositions, inferences, theories, and axioms. Each chapter consists of numbered paragraphs and subparagraphs. Not being a philosopher or logician, I do not know whether this is a standard format for such a book.
The author is a retired philosopher, who taught at the University of Düsseldorf and founded and co-edited (until 2009) the Journal for the General Philosophy of Science. This information is not included in the book, but can be gleaned from either the German Wikipedia site or amazon.com. According to the translator’s introduction, the idea that underlies this book — that logic and mathematics rest wholly upon sense experience — is an empiricist Berkeleyan thesis with few defenders. Further, “the book is not primarily intended as a treatise which explicates and defends theoretical propositions in the philosophy of logic and mathematics.” Rather, it “seeks to analyze and refashion logical elements and provide a usable ‘how-to’ manual for the practical carrying-on of an empiricist logic as an alternative to current logical practice.” According to the description on the book’s back cover, the idea of the pyramidal schema is original to the author and “offers a key innovation in its ‘pyramidal’ graph system” for the logical formalization of topics of mathematical logic, namely, concepts, definitions, inferences, theories, and axioms.
To give some flavor of the author’s style of writing, I will quote a bit. “Mathematical logic discusses the attributes of the concept of number in the terminology of set theory. ‘Set’ is, from a logical viewpoint simply another word for the logical connector which signals particularization. For that reason ‘set’, as understood in classical logic means the same as ‘some’. … One easily sees: It is not the concept of the set which explains number, but the reverse. ’Set’ is understood according to the meaning of number.” (1.15.6, pp. 22-23).
I found it hard to decipher where the author is coming from or what his thesis entails. Perhaps someone more familiar with empiricist philosophy could figure that out and gain from reading this book.
Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. In 2003, she was elected a Fellow of the American Association for the Advancement of Science. She remains active in mathematics education research and curriculum development.