Frédéric Patras.The Essence of Numbers

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This book approaches the issue of the essence of numbers from a very broad perspective: historical, philosophical, and mathematical. The author is a practicing mathematician who has been working in philosophy since the 1990s, with a particular focus on phenomenology, while maintaining his activity in mathematics. As he explains in the preface to the English translation, the original French version began with a series of lectures introducing philosophy students at Nice to the philosophy of science and of mathematics via the philosophy of arithmetic. The initial aim was to develop in the students an understanding of the issues from the origins in classical Greek thought to the point where they would be ready to follow Husserl’s Philosophy of Arithmetic, which Patras considers ‘one of the deepest texts ever written on mathematical thought’. Each chapter is a manageable chunk of about ten pages, presumably covering what was given as a single lecture. Chapter 1 is an introduction, setting up the plan for the book. While our intuitive conception of numbers appears not to have changed much since classical Greece, the development within mathematics of the theory of numbers is quite extensive. The book aims to help resolve this tension by investigating the ‘essence’ of numbers, starting with classical Greece and following the development of the concept through to the modern era with its rigorous definitions. There is then a very brief overview: starting with the Greeks and the ‘problem of the One’ — is one a number? While civilizations before ancient Greece used numbers and calculation, the Greeks were the first to consider its definition. Number, and the study of the arithmetical properties of the continuum, have been a concern of mathematics throughout its history. Developments at the end of the nineteenth century, and especially Frege’s contributions bringing ‘arithmetic back to the pure laws of thought’, were central to the development of the concept of number, ‘a decisive issue for the whole theory of knowledge’. The algebraic nature of numbers and arithmetic led to the possibility of extending beyond the natural numbers.