Logic in the History and Philosophy of Mathematical Practice

Mathematical logic is the study of reasoning about mathematical objects and the degree to which mathematical and scientific reasoning can be formalized and mechanized. Logic provides the foundations of mathematics and of theoretical computer science. Classical logic defined truth, developed the theory of infinite numbers, resolved paradoxes of naive set theory, defined what an algorithm is, and established that certain mathematical principles are independent from the rest of mathematics. Modern logic contributes to current developments in mathematics and its foundations, and continues to closely interact with various areas of mathematics. Essential and surprising connections are often found in order to solve big problems. To bring various resources into a single argument, common methods, uniform terminology, and concisely stated theorems are needed. This interaction of disciplines has not only led to new approaches and results but also to new areas of mathematical research, such as computable structure theory, reverse mathematics, and the emerging quantum computing theory.