Brouwer takes the intuition of time to belong to pre-linguistic consciousness. Mathematics, therefore, is essentially languageless. It is the activity of effecting non-linguistic constructions out of something that is not of a linguistic nature. Using language we can describe our mathematical activities, but these activities themselves do not depend on linguistic elements, and nothing that is true about mathematical constructional activities owes its truth to some linguistic fact. Linguistic objects such as axioms may serve to describe a mental construction, but they cannot bring it into being. For this reason, certain axioms from classical mathematics are rejected by intuitionists, such as the completeness axiom for real numbers, which says that if a non-empty set of real numbers has an upper bound, then it has a least upper bound: we know of no general method that would allow us to construct mentally the least upper bound whose existence the axiom claims.

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