One of the successful results of such a program is the ability to study mathematical language and reasoning using mathematics itself. For example, we will eventually give a precise mathematical definition of a formal proof, and to avoid confusion with our current intuitive understanding of what a proof is, we will call these objects deductions. One can think of our eventual definition of a deduction as analogous to the precise mathematical definition of continuity, which replaces the fuzzy “a graph that can be drawn without lifting your pencil.” Once we have codified the notion in this way, we will have turned deductions into precise mathematical objects, allowing us to prove mathematical theorems about deductions using normal mathematical reasoning. For example, we will open up the possibility of proving that there is no deduction of certain mathematical statements.
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