Brouwer's intuitionalism, Formalism and Logicism
Gödel’s incompleteness theorems were mathematically proven results but they had broad philosophical consequences. They were proofs that would show that there are certain true propositions that are improvable. They were epistemological truths, meaning they dealt with the nature of knowledge itself by proving an absolute limitation on what we can mathematical prove. (Goldstein 2013)
To assess the effects of Gödel’s results, the theorems themselves will be outlined, as will the three schools of logicism, formalism and intuitionism, then the effects of the theorems on the schools shall be considered. To appreciate the consequences of the incompleteness theorems there is a need to explain the key terms of consistency and completeness and
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This view limits meaningful mathematics to any process that has a finite number of steps is permissible (Brown 2008, 127). This meant that Brouwer also rejected any classical mathematics that depended on proof by contradiction as a ‘meaningless combination of words’ (Snapper 1979). Hilbert agreed with Brouwer that the completed infinite went ‘beyond intuitive evidence. But refused to follow Brouwer in giving up classical mathematics…’ (Kleene 1952, 53). According to Weyl 1946, Brouwer had the belief that this absolute infinite in classical mathematics ‘transcends the human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence’ (Kleene 1952, 49).
Gödel’s results seem to verify Intuitionism’s criticisms about classical mathematics. However they also highlight the confusing nature of the intuitionistic notion of provability, with which intuitionists equate truth. (Raatikainen 2005). Gödel showed that provability could never be inexplicably linked to truth like this. In intuitionism, the notions of ‘provability and constructivity are vague and indefinite and lack complete perspicuity and clarity’ (Raatikainen 2005).
Its limited scope allowed it to be less affected by the incompleteness theorems then the other schools. As if we take ‘p’ to be either true or not true, then intuitionists would identify the