 Philosophically,  intuitionism  differs from  logicism  by treating logic as a part of mathematics rather than as the foundation of mathematics; from  -->finitism-->  by allowing constructive reasoning about potentially uncountable structures (e.g., monotone bar induction on the tree of potentially infinite sequences of natural numbers); and from  Platonism  by viewing mathematical objects as mental constructs with no independent ideal existence. Hilbert’s  formalist program,  to justify classical mathematics by reducing it to a formal system whose consistency should be established by finitistic (hence constructive) means, was the most powerful contemporary rival to Brouwer’s developing intuitionism. In his 1912 essay Intuitionism and Formalism Brouwer correctly predicted that any attempt to prove the consistency of complete induction on the natural numbers would lead to a vicious circle. 