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Discussions of the foundations of mathematics and their history are frequently restricted to logical issues in a narrow sense, or else to traditional problems of analytic philosophy. The viewpoints that serve this purpose included the foundational ideas of working mathematicians, such as Kronecker, Dedekind, Borel and the early Hilbert, and the development of notions like model and modelling, arbitrary function, completeness, and non-Archimedean structures. The philosophers discussed include not only the household names in logic, but also Husserl, Wittgenstein and Ramsey. Jaakko Hintikka is the author or co-author of thirty volumes and of some scholarly articles in mathematical and philosophical logic, epistemology, language theory, philosophy of science, history of ideas and history of philosophy, including Aristotle, Descartes, Leibniz, Kant, Peirce, The Bloomsbury Group, Husserl and Wittgenstein.

Please wait Description : This volume contains the two most important essays on the logical foundations of the number system by the famous German mathematician J. The first presents Dedekind's theory of the irrational number-the Dedekind cut idea-perhaps the most famous of several such theories created in the 19th century to give a precise meaning to irrational numbers, which had been used on an intuitive basis since Greek times. This paper provided a purely arithmetic and perfectly rigorous foundation for the irrational numbers and thereby a rigorous meaning of continuity in analysis. The second essay is an attempt to give a logical basis for transfinite numbers and properties of the natural numbers.

A Dedekind cut is a partition of the rational numbers into two sets A and B , such that all elements of A are less than all elements of B , and A contains no greatest element. The set B may or may not have a smallest element among the rationals. If B has a smallest element among the rationals, the cut corresponds to that rational.

Julius Wilhelm Richard Dedekind 6 October — 12 February was a German mathematician who made important contributions to number theory , abstract algebra particularly ring theory , and the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as Logicism.

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