science: The Abstraction of Mathematics

In mathematics a movement toward the abstract, axiomatic approach began early in the 19th cent. with the discovery of two different types of non-Euclidean geometries and various abstract algebras, some of them noncommutative. While there has been a tendency to consolidate and unify under a few general concepts, such as those of group, set, and transformation, there has also been considerable research in the foundations of mathematics, with a close examination of the nature of these and other concepts and of the logical systems underlying mathematics.

Sections in this article:

• Introduction
• Promise and Problems of Modern Science
• Modern Science and Technology
• Astronomy beyond the Visual Spectrum
• The Abstraction of Mathematics
• Biology Becomes an Interdisciplinary Science
• Advances in Chemistry
• Quantum Theory and the Theory of Relativity
• The Impact of Elementary Particles
• Revolutions in Modern Science
• Science and the Industrial Revolution
• New Ideas in Biology
• Birth of Modern Geology
• Advances in Astronomy
• Innovations in Chemistry
• The Evolution of Mathematics and Physics
• The Age of Classical Science
• The Impact of Sir Isaac Newton
• Improved Communication of Scientific Knowledge
• The Rejection of Traditional Paradigms
• The Scientific Revolution
• The Craft Tradition and Early Empiricism in Europe
• Muslim Preservation of Learning
• Scientific Progress in China and India
• Influence of the Alexandrian Schools
• Early Greek Contributions to Science
• Practical Applications in the Ancient Middle East
• The Beginnings of Science
• Branches of Specialization
• Role of Measurement and Experiment
• The Scientific Method
• Bibliography

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