As long as algebra and geometry The unreasonable effectiveness of proceeded along separate paths, mathematics in science . . . Eugene Wigner their advance was slow and their applications limited. But when these sciences joined Weil, if you knows of a better 'oie, company, they drew from each go to it. Bruce Bairnsfather other fresh vitality and thence forward marched on at a rapid pace What is now proved was once only towards perfeetion. imagined. Wi1liam Blake J oseph Louis Lagrange Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. This series of books, Mathematics and Its Applications, is devoted to such (new) interrelations as exempla gratia: - a central concept which plays an important role in several different mathematical and/or scientific specialized areas; Editor's Preface 8 - new applications of the results and ideas from one area of scientific endeavor into another; - influences which the results, problems and concepts of one field of inquiry have and have had on the development of another. With books on topics such as these, of moderate length and price, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practitioners in diversified fields.
I. Semantical and Syntactical Aspects of Elementary Mathematical Theories.- I.1. Introduction to the Elementary Predicate Calculus without Equality.- I.2. Semantical Interpretation of the Propositional Calculus.- I.3. Semantical Interpretation of the Elementary Predicate Calculus.- I.4. Decision Procedure for the Elementary Predicate Calculus.- I.5. Predicate Calculus - the Theory Z.- I.6. Godel's Incompleteness Theorem.- I.7. The Incompleteness Theorems and Semantics.- I.8. Remarks on Non-Standard Mathematics.- II. Epistemological Aspects of Mathematics in Historical Perspective.- II. 1. Introduction.- II.2. The Philosophy of Mathematics in History.- II.2.1 Greek Mathematics.- II.2.2. From Hellenistic Philosophy to Modern Rationalism.- II.2.3. The Period in Early Modern Philosophy.- II.2.3.1. Descartes (1596-1650).- II.2.3.2. Newton (1642-1727).- II.2.3.3. Leibniz (1646-1716).- II.2.3.4. Kant (1724-1804).- II.3. Transition to the Present Century.- II.4. Directions in the 20th Century Philosophy of Mathematics.- II.4.1. Logicism.- II.4.1.1. Frege (1848-1925).- II.4.1.2. Russell (1872-1970).- II.4.2. Intuitionism.- II.4.2.1. Poincare (1854-1912).- II.4.2.2. Brouwer (1881-1966).- II.4.2.3. The 'Bourbaki' Group.- II.4.3. Formalism.- III. An Outline of a Complementaristic Approach to Mathematics.- III.1. Facets and Methods of a Philosophy of Mathematics.- III.2. Two Kinds of Mathematical Existence.- III.3. Language, Set Theory and Mathematical Complementarity.- III.4. Complementarist Set Theory - an Outline.- III.5. The Unity of Mathematics: Algebra and Topology.- III.6. Bridging the Abyss Between the Discrete and the Continuous.- Selected Bibliography.- References for Further Study.- Index of Names.- Index of Subjects.