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What is a brief explanation of formalistic school of thought?

The term ‘formalism’ is ambiguous many times over.

On one of its disambiguations, it refers to the doctrine that mathematical assertions, when true, are true strictly in virtue of their formal properties.

This particular doctrine proved to be false, since, as it turns out, there is no purely formal definition of formal truth. See Proof of the Incompleteness of Deductive Logic

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