Metamathematics
Metamathematics, sometimes called metalogic, is the branch of logic concerning the combination and application of mathematical symbols. The primary goal of metamathematics is to determine the nature of mathematical reasoning. It consists of basic principles, mainly concerned with proofs of consistency that attempt to formulate mathematical theories. Metamathematics today is mainly used as a mechanism of proving mathematical theories in an automated way.
Metamathematics arose during the attempts in the late 1800s to mechanize the verification of mathematical proofs. At that time precise mathematical logic coexisted on equal terms with vague intuitions and it was becoming apparent that a better, more firm basis for mathematics be developed. From the late 19th century studies in formal logic attempted to develop a complete, consistent formulation of mathematics such that proposals could be formally stated and proved or disproved using a limited number of symbols that had well defined meanings. Theorems, formulated to follow strict rules of inference, emerged from axioms like branches of a tree. The axioms were the primordial seeds from which all other things evolved. It was thought that if a systematic way of mathematics was formulated then all need for thought or judgment would be eliminated. The only requirements were that the axioms were true statements and the rules of inference were truth preserving. This, it was thought, would lead to mathematics in which falsehoods simply could not be present.
Although statements were formally written using standard numerals, arithmetic signs, parentheses and so on it was thought that this notation was not a necessary feature. The statements could be built out of any arbitrary set of symbols as long as they were consistent and defined. This was of looking at mathematics was novel and named metamathematics. The difficulties in this task started to become realized in 1925 when Whitehead and Russell published Principia Mathematica. Hundreds of pages of symbols were required before the simple statement 1 + 1 = 2 could be deduced. In 1931 the realization of the impossible feat of developing such a formulation was fully realized as Gödel's proof of his incompleteness theorem showed that nearly a century of efforts by the world's greatest mathematicians was doomed to failure. He determined that by thinking of theorems as patterns of symbols that it is possible for a statement in a formal system not only to talk about itself but to also deny its own theoremhood. Thus Gödel showed that this formal system would not capture all true statements of mathematics. All formal systems are incomplete because they are able to express statements that say of themselves that they are unprovable. It is not math itself that is incomplete but any formal system that attempts to capture all the truths of mathematics in a finite set of axioms and rules that is incomplete. Gödel was 25 when he developed the incompleteness theorem. Although the proof of the theorem removed any possibility of mechanizing the verification of mathematical proofs in a formal system it also did something else. The article in which the incompleteness theorem was published invented the theory of recursive functions, which today is the basis of a powerful theory of computing.
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