The Foundational Crisis of Mathematics
The need for a formal definition of the concept of algorithm was made clear during the first few decades of the twentieth century as a result of events taking place in mathematics. At the beginning of the century, Cantor’s naive set theory was born. This theory was very promising because it offered a common foundation to all the fields of mathematics. However, it treated infinity incautiously and boldly. This called for a response, which soon came in the form of logical paradoxes. Because Cantor’s set theory was unable to eliminate them—or at least bring them under control—formal logic was engaged. As a result, three schools of mathematical thought—intuitionism, logicism, and formalism—contributed important ideas and tools that enabled an exact and concise mathematical expression and brought rigor to mathematical research.
KeywordsNatural Number Mathematical Object Basic Notion Axiomatic System Classical Mathematic
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