Metamathematics

Emphasis on metamathematics (and perhaps the creation of the term itself) owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century.

In his hands, it meant something akin to contemporary proof theory, in which finitary methods are used to study various axiomatized mathematical theorems (Kleene 1952, p. 55).

Other prominent figures in the field include Bertrand Russell, Thoralf Skolem, Emil Post, Alonzo Church, Alan Turing, Stephen Kleene, Willard Quine, Paul Benacerraf, Hilary Putnam, Gregory Chaitin, Alfred Tarski, Paul Cohen and Kurt Gödel.

When Gauss discovered hyperbolic geometry, it is said that he did not publish anything about it out of fear of the "uproar of the Boeotians", which would ruin his status as princeps mathematicorum (Latin, "the Prince of Mathematicians").

[1] The "uproar of the Boeotians" came and went, and gave an impetus to metamathematics and great improvements in mathematical rigour, analytical philosophy and logic.

Principia Mathematica, or "PM" as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven.

As such, this ambitious project is of great importance in the history of mathematics and philosophy,[2] being one of the foremost products of the belief that such an undertaking may be achievable.

One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets.

T-theories form the basis of much fundamental work in philosophical logic, where they are applied in several important controversies in analytic philosophy.