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Concise, accessible sketches of the views of Plato, Aristotle, Leibniz, and Kant highlight this study of the general structure and foundation of pure and applied mathematics. Author Stephan K#65533;rner dedicates two chapters apiece -- one expository and one critical -- to each of the three main modern schools of thought on mathematical philosophy: the formalists, the logicists, and the intuitionists. After critically examining the propositions and theories of each philosophy, K#65533;rner presents a new position concerning the relation between mathematical theories, empirical data, and philosophical presuppositions. 
The Review of Metaphysics praised this volume as "a lucid and stimulating essay which combines accuracy and sophistication with a minimum of technical language." Compact but comprehensive, this nontechnical introduction will appeal to professionals, students, and other readers interested in the intersection of philosophical problems with pure and applied mathematics.

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• Preface
• Introduction
• I Some older views
• 1 Plato's account
• 2 Some views of Aristotle
• 3 Leibniz's philosophy of mathematics
• 4 Kant: some of his views
• II Mathematics as Logic: Exposition
• 1 The programme
• 2 The logic of truth-functions
• 3 On the logic of classes
• 4 On the logic of quantification
• 5 On logicist systems
• III Mathematics as Logic: Criticism
• 1 The ligicist account of logic
• 2 The logist conflation of empirical and non-empirical concepts
• 3 The logicist theory of mathematical infinity
• 4 The logicist account of geometry
• IV Mathematics as the Science of Formal Systems: Exposition
• 1 The programme
• 2 Finite methods and infinite totalities
• 3 Formal systems and formalizations
• 4 Some results of metamathematics
• V Mathematics as the Science of Formal Systems: Criticism
• 1 The formalist account of pure mathematics
• 2 The formalist account of applied mathematics
• 3 The concept of actual infinity
• 4 The formalist conception of logic
• VI Mathematics as the Activity of Intuitive Constructions: Exposition
• 1 The programme
• 2 Intuitionist mathematics
• 3 Intuitionist logic
• VII Mathematics as the Activity of Intuitive Constructions: Criticism
• 1 Mathematical theorems as reports on intuitive constructions
• 2 Intuitionism and the logical status of applied mathematics
• 3 The intuitionist conception of mathematical infinity
• 4 Interrelations between formalism and intuitionism
• VIII The nature of pure and applied mathematics
• 1 Exact and inexact concepts
• 2 Pure mathematics disconnected from perception
• 3 Mathematical existence-propositions
• 4 The nature of applied mathematics
• 5 Mathematics and philosophy
• Appendix A On the classical theory of real numbers
• Appendix B Some suggestions for further reading
• Index