Brouwer's intuitionalism, Formalism and Logicism Essay

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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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(Struik 1987, 203). Logicism disagrees with Intuitionism as it asserts that we do not create knowledge but simply reveal existing truths (Brown 2008, 125). Since in intuitionism ‘abstract entities are admitted only if they are man made’ (Snapper 1979, 209). Brouwer’s criticism of Logicism is that they use the principles of finite sets and their subsets as a form of logic beyond and prior to mathematics and used it to reason about infinite sets (Kleene 1952, 46-7).

Intuitionism was developed as a reaction to Cantor’s set theory and its paradoxes. Intuitionists sought to rebuild mathematics from the ‘bottom-up’. They saw Mathematics as ‘an activity’; Mathematicians do not access pre-existing knowledge but construct knowledge (Brown 2008, 121).

Brouwer saw logic as an unreliable basis for mathematics and therefore Brouwer’s intuitionism sees mathematics as having its foundations with ‘Ur-Intuition, a basic intuition of the natural numbers’ (Struik 1987, 202, Palmgren 2009). Its fundamental and defining characteristic is its analysis of what it means for a statement to be true. In Brouwer’s original intuitionism he demands ‘truth though constructivity’ (Struik 1987, 202). This means that he only allowed entities that had a clear and definable method of construction. For example, in this way Brouwer would accept the idea of possible infinitely, as it required a continuous set of constructions and ‘remains forever in the

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