Kurt Gödel and the Mechanization of Mathematics
&gt; formalism: the idea that mathematics can be reconstructed in a content-free manner, or more strongly put, that mathematics itself is nothing but “a formal game of symbols”.<p>This is a common misrepresentation of Hilbert&#x27;s Formalism. It doesn&#x27;t state that <i>no</i> mathematical sentence has content beyond its mere symbols, but that <i>not every</i> mathematical sentence has content beyond its mere symbols.<p>Formalism was meant as a compromise between the two dueling philosophies of mathematics of the time: Russel&#x27;s Logicism, which essentially said that all of mathematics can be derived from some self-evident, obviously true, logical axioms, and Brouwer&#x27;s Intuitionism, which essentially said that the truth of mathematics stems from empirical confirmation that breaks down when infinities are involved, and therefore some logical axioms cannot be taken to be true for infinites. Hilbert tried to accept Brouwer&#x27;s finitist philosophical position without rejecting any useful axioms. Formalism, then, separates mathematical sentences into two kinds: &quot;real&quot; -- those dealing with finite quantities and can be tied to physical observation and prediction, and &quot;ideal&quot; -- those dealing with infinities. Instead of rejecting the latter, Formalism says that as long as they don&#x27;t introduce inconsistencies that invalidate the former, one could view them -- the <i>ideal</i> sentences -- as having no content beyond symbols, i.e. as not referring to anything &quot;real&quot;.