Formalism is a mathematical school of thought that was headed by the German mathematician David Hilbert. Formalists argue that mathematics must be developed through axiomatic systems. Formalists agree with Platonism on the principles of mathematical proof, but Hilbert's followers don't recognize an external world of mathematics. Formalists argue that mathematical objects don't exist until we define them. Humans create the real number system, for example, by establishing axioms to describe it. All that mathematics needs are inference rules to progress from one step to the next. Formalists tried to prove that within the framework of established axioms, theorems, and definitions, a mathematical system is consistent and, in the mid-20th century, formalism became the predominant philosophical attitude in math textbooks. However, it was undermined by Gödel's incompleteness theorem and also the general recognition that results can be usefully applied without having to be proved or derived axiomatically.
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