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# The Importance of Formalizing the Meta-Mathematics

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 universality20 This gave rise to the claim that inductive considerations can show no more than that the axioms are plausible or probably true20 which necessitated another approach Hilbert proposed a method of attaining absolute proofs through a complete formalization of a deductive system26 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 clarity27 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 on27 What is meant by this analogy is that meta-mathematics isnt 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 mathematics the description discussion and theorizing about the systems belong in the file marked meta-mathematics32 By using meta-mathematics Hilbert hoped to show by exhaustively examining these

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