Foundations of Mathematics - Foundational Crisis

Foundational Crisis

The foundational crisis of mathematics (in German: Grundlagenkrise der Mathematik) was the early 20th century's term for the search for proper foundations of mathematics.

Several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, as the assumption that mathematics had any foundation that could be consistently stated within mathematics itself was heavily challenged by the discovery of various paradoxes (such as Russell's paradox).

The name "paradox" should not be confused with contradiction. A contradiction in a formal theory is a formal proof of an absurdity inside the theory (such as 2 + 2 = 5), showing that this theory is inconsistent and must be rejected. But a paradox may either refer to a surprising but true result in a given formal theory, or to an informal argument leading to a contradiction, so that a candidate theory where a formalization of the argument might be attempted must disallow at least one of its steps; in this case the problem is to find a satisfying theory without contradiction. Both meanings may apply if the formalized version of the argument forms the proof of a surprising truth. For instance, Russell's paradox may be expressed as "there is no set of all sets" (except in some marginal axiomatic set theories).

Various schools of thought on the right approach to the foundations of mathematics were fiercely opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which thought to ground mathematics on a small basis of a logical system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols (van Dalen, 2008). The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time.