Question

In about 250 words discuss the history of foundations of mathematics as proposed in the late 19^th and early 20^th centuries.

Answer 1

Foundations of mathematics is the study of the philosophical and logical[1] and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.

Entire branches of mathematics have been created in the 19th and 20th centuries including non-Euclidean geometry, trans-finite arithmetic, rigorous set theory, group theory, topology, category theory, chaos theory, and we are just scratching the surface.Few breakthroughs in Mathematics make it into the public consciousness until the underlying theory is used in some practical way. Chaos Theory has been popularised by the Butterfly Effect. Group theory is know for being used in particle physics and in Rubik's cube. Non-Euclidian geometry is used for space-time in the general Theory of Relativity.

The systematic search for the foundations of mathematics started at the end of the 19th century and formed a new mathematical discipline called mathematical logic, with strong links to theoretical computer science. It went through a series of crises with paradoxical results, until the discoveries stabilized during the 20th century as a large and coherent body of mathematical knowledge with several aspects or components (set theory, model theory, proof theory, etc.), whose detailed properties and possible variants are still an active research field. Its high level of technical sophistication inspired many philosophers to conjecture that it can serve as a model or pattern for the foundations of other sciences.