Meta-Mathematics: An exploration of Godels Incompleteness Theorems

The idea that mathematics could contain inherent contradictions acquired much speculation and criticism as it put into question the fundamental system by which we interpret the world. In searching for mathematical proofs that are consistent and hold no contradictions, Hilbert tried different methods one of which involved using models, but this proved logically incomplete, "for even if all the observed facts are in agreement with the axioms, the possibility is open that a hitherto unobserved fact may contradict them and so destroy their title to universality"(20). This gave rise to the claim that "inductive considerations can show no more than that the axioms are plausible or probably true"(20), which necessitated another approach.

Hilbert proposed a method of attaining absolute proofs through a "complete formalization of a deductive system"(26); these were the building blocks of a new system called 'meta-mathematics'. Meta-mathematics was intended to formalize mathematics. Formalizing a system, such as mathematics is beneficial because "it reveals structure and function in naked clarity"(27).

Mathematics..

Saarbrücken, HTW, Mathematics Workshop

Mathematics Pond

This formalization would allow mathematicians to see all the structural patterns, and relations between all the symbols and equations of math. It is in this way that meta-mathematics is an explanation, or a deconstruction of mathematics.

To explain the meaning of meta-mathematics Nagel and Newman use an analogy explaining that, "one may say that a 'string' is pretty, or that it resembles another 'string', or that one 'string' appears to be made up of three others, and so on"(27). What is meant by this analogy is that meta-mathematics isn't mathematics, it is a formalized system used to describe mathematics. It makes comments about mathematics, just like in the analogy where one would be making comments about the string. The clearest distinction drawn between mathematics and meta-mathematics is, "formal systems that mathematicians construct belong in the file labeled...