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  "body": "\n## Definition\n\n*Formalism* is a position in the [[foundations of mathematics|EFFRJFW67Y]].\nIt holds that mathematical statements are strings of symbols in a formal system.\nA statement is established when it is derived from the axioms of the system by its rules of inference.\nThe derivation is the whole content of a [[proof|0B94TKFQJX]], and a statement need not refer to abstract objects.\n\n## Remarks\n\nIn a formal system for arithmetic, the string $S(0) + S(0) = S(S(0))$ is established by a derivation from the axioms; a formalist accepts it for that reason alone.\nSome statements are true in the standard natural numbers but have no derivation in the system.\nAn example is a Gödel sentence of the [[incompleteness theorems|X9WYNYE29C]], and formalism does not count it as established.\nHilbert's version of formalism kept a contentful core of finitary mathematics.\nIt treated the rest as a formal system whose consistency should be proved by finitary means; this is [[Hilbert's program|P1RW6JWX73]].\n",
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